Shapely Documentation
Release 1.8.1

Sean Gillies

Feb 16, 2022

CONTENTS

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Documentation Contents
1.1 Shapely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Shapely User Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Migrating to Shapely 1.8 / 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
26
79

2

Indices and tables

87

Index

89

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CHAPTERONE

DOCUMENTATION CONTENTS

1.1 Shapely
Manipulation and analysis of geometric objects in the Cartesian plane.

Shapely is a BSD-licensed Python package for manipulation and analysis of planar geometric objects. It is based on
the widely deployed GEOS (the engine of PostGIS) and JTS (from which GEOS is ported) libraries. Shapely is notconcerned with data formats or coordinate systems, but can be readily integrated with packages that are. For moredetails, see:
• Shapely GitHub repository• Shapely documentation and manual1Shapely Documentation, Release 1.8.1

1.1.1 Usage
Here is the canonical example of building an approximately circular patch by buffering a point.
>>> from shapely.geometry import Point>>> patch = Point(0.0, 0.0).buffer(10.0)>>> patch<shapely.geometry.polygon.Polygon object at 0x...>>>> patch.area
313.65484905459385
See the manual for more examples and guidance.

1.1.2 Requirements
Shapely 1.8 requires• Python >=3.6• GEOS >=3.3

1.1.3 Installing Shapely
Shapely may be installed from a source distribution or one of several kinds of built distribution.
Built distributions
Built distributions are the only option for users who do not have or do not know how to use their platform’s compiler
and Python SDK, and a good option for users who would rather not bother.
Linux, OS X, and Windows users can get Shapely wheels with GEOS included from the Python Package Index with arecent version of pip (8+):$ pip install shapely
Shapely is available via system package management tools like apt, yum, and Homebrew, and is also provided bypopular Python distributions like Canopy and Anaconda. If you use the Conda package manager to install Shapely, besure to use the conda-forge channel.
Windows users have another good installation options: the wheels published at https://www.lfd.uci.edu/~gohlke/
pythonlibs/#shapely. These can be installed using pip by specifying the entire URL.
Source distributions
If you want to build Shapely from source for compatibility with other modules that depend on GEOS (such as cartopy
or osgeo.ogr) or want to use a different version of GEOS than the one included in the project wheels you should first
install the GEOS library, Cython, and Numpy on your system (using apt, yum, brew, or other means) and then direct
pip to ignore the binary wheels.
$ pip install shapely --no-binary shapely
If you’ve installed GEOS to a standard location, the geos-config program will be used to get compiler and linker options.
If geos-config is not on your executable, it can be specified with a GEOS_CONFIG environment variable, e.g.:2

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Shapely Documentation, Release 1.8.1$ GEOS_CONFIG=/path/to/geos-config pip install shapely

1.1.4 Integration
Shapely does not read or write data files, but it can serialize and deserialize using several well known formats andprotocols. The shapely.wkb and shapely.wkt modules provide dumpers and loaders inspired by Python’s pickle module.
>>> from shapely.wkt import dumps, loads>>> dumps(loads('POINT (0 0)'))'POINT (0.0000000000000000 0.0000000000000000)'
Shapely can also integrate with other Python GIS packages using GeoJSON-like dicts.
>>> import json>>> from shapely.geometry import mapping, shape>>> s = shape(json.loads('{"type": "Point", "coordinates": [0.0, 0.0]}'))>>> s<shapely.geometry.point.Point object at 0x...>>>> print(json.dumps(mapping(s))){"type": "Point", "coordinates": [0.0, 0.0]}

1.1.5 Development and Testing
Dependencies for developing Shapely are listed in requirements-dev.txt. Cython and Numpy are not required for production installations, only for development. Use of a virtual environment is strongly recommended.
$ virtualenv .
$ source bin/activate(env)$ pip install -r requirements-dev.txt(env)$ pip install -e .
The project uses pytest to run Shapely’s suite of unittests and doctests.
(env)$ python -m pytest

1.1.6 Support
Questions about using Shapely may be asked on the GIS StackExchange using the “shapely” tag.
Bugs may be reported at https://github.com/shapely/shapely/issues.

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1.1.7 Credits
Shapely is written by:• Adi Shavit <adishavit@gmail.com>• Alberto Rubiales <arubiales11@gmail.com>• Allan Adair <allan.m.adair@gmail.com>• Andrew Blakey <ablakey@gmail.com>• Andy Freeland <andy@andyfreeland.net>• Ariel Kadouri <ariel@arielsartistry.com>• Aron Bierbaum <aronbierbaum@gmail.com>• Bart Broere <2715782+bartbroere@users.noreply.github.com>• Bas Couwenberg <sebastic@xs4all.nl>• Ben Beasley <code@musicinmybrain.net>• Benjamin Root <ben.v.root@gmail.com>• BertrandGervais <bertrand.gervais.pro@gmail.com>• Bhavika Tekwani <4955119+bhavika@users.noreply.github.com>• Bi0T1N <Bi0T1N@users.noreply.github.com>• Brad Hards <bradh@frogmouth.net>• Brandon Wood <btwood@geometeor.com>• Chad Hawkins <cwh@chadwhawkins.com>• Christian Prior <cprior@gmail.com>• Christian Quest <github@cquest.org>• Christophe Pradal <christophe.pradal@inria.fr>• Dan Baston <dbaston@gmail.com>• Dan Mahr <danmahr23@gmail.com>• Daniele Esposti <expobrain@users.noreply.github.com>• Dave Collins <dave@hopest.net>• David Baumgold <david@davidbaumgold.com>• David Swinkels <davidswinkelss@gmail.com>• Denis Rykov <rykovd@gmail.com>• Erwin Sterrenburg <e.w.sterrenburg@gmail.com>• Felix Divo <4403130+felixdivo@users.noreply.github.com>• Felix Yan <felixonmars@archlinux.org>• Filipe Fernandes <ocefpaf@gmail.com>• Frédéric Junod <frederic.junod@camptocamp.com>• Gabi Davar <grizzly.nyo@gmail.com>• Gerrit Holl <gerrit.holl@dwd.de>4

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Shapely Documentation, Release 1.8.1• Hannes <kannes@users.noreply.github.com>• Hao Zheng <Furioushaozheng@gmail.com>• Henry Walshaw <henry.walshaw@gmail.com>• Howard Butler <hobu.inc@gmail.com>• Hugo <hugovk@users.noreply.github.com>• Jacob Wasserman <jwasserman@gmail.com>• Jaeha Lee <jaehaaheaj@gmail.com>• James Douglass <jamesdouglassusa@gmail.com>• James Gaboardi <jgaboardi@gmail.com>• James Lamb <jaylamb20@gmail.com>• James McBride <jdmcbr@gmail.com>• James Spencer <james.s.spencer@gmail.com>• Jamie Hall <jamie1212@gmail.com>• Jason Sanford <jason.sanford@mapmyfitness.com>• Jeethu Rao <jeethu@jeethurao.com>• Jeremiah England <34973839+Jeremiah-England@users.noreply.github.com>• Jinkun Wang <mejkunw@gmail.com>• Johan Euphrosine <proppy@aminche.com>• Johannes Schönberger <jschoenberger@demuc.de>• Jonathan Schoonhoven <jschoonhoven@lyft.com>• Joris Van den Bossche <jorisvandenbossche@gmail.com>• Joshua Arnott <josh@snorfalorpagus.net>• Juan Luis Cano Rodríguez <juanlu@satellogic.com>• Justin Shenk <shenk.justin@gmail.com>• Kai Lautaportti <dokai@b426a367-1105-0410-b9ff-cdf4ab011145>• Kelsey Jordahl <kjordahl@enthought.com>• Kevin Wurster <wursterk@gmail.com>• Konstantin Veretennicov <kveretennicov@gmail.com>• Koshy Thomas <koshy1123@gmail.com>• Kristian Evers <kristianevers@gmail.com>• Kyle Barron <kylebarron2@gmail.com>• Leandro Lima <leandro@limaesilva.com.br>• Lukasz <uhho@users.noreply.github.com>• Luke Lee <durdenmisc@gmail.com>• Maarten Vermeyen <maarten.vermeyen@rwo.vlaanderen.be>• Marc Jansen <jansen@terrestris.de>

1.1. Shapely

5Shapely Documentation, Release 1.8.1• Marco De Nadai <me@marcodena.it>• Mathieu <mathieu.nivel@gmail.com>• Matt Amos <matt.amos@mapzen.com>• Matthias Cuntz <mcuntz@users.noreply.github.com>• MejstrikRudolf <68251685+MejstrikRudolf@users.noreply.github.com>• Michael K <michael-k@users.noreply.github.com>• Michel Blancard <michel.blancard@data.gouv.fr>• Mike Taves <mwtoews@gmail.com>• Morris Tweed <tweed.morris@gmail.com>• Naveen Michaud-Agrawal <naveen.michaudagrawal@gmail.com>• Oliver Tonnhofer <olt@bogosoft.com>• Paveł Tyślacki <tbicr@users.noreply.github.com>• Peter Sagerson <psagers.github@ignorare.net>• Phil Elson <pelson.pub@gmail.com>• Pierre PACI <villerupt@gmail.com>• Raja Gangopadhya <raja.gangopadhya@ridewithvia.com>• Ricardo Zilleruelo <51384295+zetaatlyft@users.noreply.github.com>• Rémy Phelipot <remy-phelipot@users.noreply.github.com>• S Murthy <sr-murthy@users.noreply.github.com>• Sampo Syrjanen <sampo.syrjanen@here.com>• Samuel Chin <samuelchin91@gmail.com>• Sean Gillies <sean.gillies@gmail.com>• Sobolev Nikita <mail@sobolevn.me>• Stephan Hügel <urschrei@gmail.com>• Steve M. Kim <steve@climate.com>• Taro Matsuzawa aka. btm <btm@tech.email.ne.jp>• Thibault Deutsch <thibault.deutsch@gmail.com>• Thomas Gratier <thomas_gratier@yahoo.fr>• Thomas Kluyver <takowl@gmail.com>• Tim Gates <tim.gates@iress.com>• Tobias Sauerwein <tobias.sauerwein@camptocamp.com>• Tom Caruso <carusot42@gmail.com>• Tom Clancy <17627475+clncy@users.noreply.github.com>• WANG Aiyong <gepcelway@gmail.com>• Will May <williamcmay@live.com>• Zachary Ware <zachary.ware@gmail.com>

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Shapely Documentation, Release 1.8.1• aharfoot <aharfoot@users.noreply.github.com>• bstadlbauer <11799671+bstadlbauer@users.noreply.github.com>• cclauss <cclauss@me.com>• clefrks <33859587+clefrks@users.noreply.github.com>• davidh-ssec <david.hoese@ssec.wisc.edu>• georgeouzou <geothrock@gmail.com>• giumas <gmasetti@ccom.unh.edu>• joelostblom <joelostblom@users.noreply.github.com>• ljwolf <levi.john.wolf@gmail.com>• mindw <grizzly.nyo@gmail.com>• rsmb <rsmb@users.noreply.github.com>• shongololo <garethsimons@me.com>• solarjoe <walterwhite666@googlemail.com>• sshuair <sshuair@gmail.com>• stephenworsley <49274989+stephenworsley@users.noreply.github.com>
See also: https://github.com/shapely/shapely/graphs/contributors.
Additional help from:• Justin Bronn (GeoDjango) for ctypes inspiration• Martin Davis (JTS)• Sandro Santilli, Mateusz Loskot, Paul Ramsey, et al (GEOS Project)
Major portions of this work were supported by a grant (for Pleiades) from the U.S. National Endowment for the Humanities (https://www.neh.gov).

1.1.8 Changes
1.8.1 (2022-02-16)
Packaging:
Wheels for 1.8.1 published on PyPI include GEOS 3.10.2. This version is the best version of GDAL yet. Discrepancies
in behavior compared to previous versions are considered to be improvements.
For the first time, we will publish wheels for macos_arm64 (see PR #1310).
Python version support:
Shapely 1.8.1 works with Pythons 3.6-3.10.
Bug fixes:• Require Cython >= 0.29.24 to support Python 3.10 (#1224).
• Fix array_interface_base (#1235).

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1.8.0 (2021-10-25)
This is the final 1.8.0 release. There have been no changes since 1.8rc2.
1.8rc2 (2021-10-19)
Build:A pyproject.toml file has been added to specify build dependencies for the _vectorized and _speedups modules (#1128).
To install shapely without these build dependencies, use the features of your build tool that disable PEP 517 and 518
support.
Bug fixes:• Part of PR #1042, which added a new primary GEOS library name to be searched for, has been reverted by PR#1201.
1.8rc1 (2021-10-04)
Deprecations:
The almost_exact() method of BaseGeometry has been deprecated. It is confusing and will be removed in 2.0.0. The
equals_exact() method is to be used instead.
Bug fixes:• We ensure that the _speedups module is always imported before _vectorized to avoid an unexplained condition
on Windows with Python 3.8 and 3.9 (#1184).
1.8a3 (2021-08-24)
Deprecations:
The STRtree class deprecation warnings have been removed. The class in 2.0.0 will be backwards compatible with theclass in 1.8.0.
Bug fixes:• The __array_interface__ raises only AttributeError, all other exceptions are deprecated starting with Numpy 1.21(#1173).
• The STRtree class now uses a pair of item, geom sequences internally instead of a dict (#1177).
1.8a2 (2021-07-15)
Python version support:
Shapely 1.8 will support only Python versions >= 3.6.
New features:• The STRtree nearest*() methods now take an optional argument that specifies exclusion of the input geometry
from results (#1115).
• A GeometryTypeError has been added to shapely.errors and is consistently raised instead of TypeError or ValueError as in version 1.7. For backwards compatibility, the new exception will derive from TypeError and Valueerror until version 2.0 (#1099).

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Shapely Documentation, Release 1.8.1• The STRtree class constructor now takes an optional second argument, a sequence of objects to be stored in thetree. If not provided, the sequence indices of the geometries will be stored, as before (#1112).
• The STRtree class has new query_geoms(), query_items(), nearest_geom(), and nearest_item() methods (#1112).
The query() and nearest() methods remain as aliases for query_geoms() and nearest_geom().
Bug fixes:• We no longer attempt to load libc to get the free function on Linux, but get it from the global symbol table.
• GEOS error messages printed when GEOS_getCoordSeq() is passed an empty geometry are avoided by neverpassing an empty geometry (#1134).
• Python’s builtin super() is now used only as described in PEP 3135 (#1109).
• Only load conda GEOS dll if it exists (on Windows) (#1108).
• Add /opt/homebrew/lib to the list of directories to be searched for the GEOS shared library.
• Added new library search path to assist app creation with cx_Freeze.
1.8a1 (2021-03-03)
Shapely 1.8.0 will be a transitional version. There are a few bug fixes and new features, but it is mainly about warning of
the upcoming changes in 2.0.0. Several more pre-releases before 1.8.0 are expected. See the migration guide to Shapely
1.8 / 2.0 for more details on how to update your code (https://shapely.readthedocs.io/en/latest/migration.html).
Python version support:
Shapely 1.8 will support only Python versions >= 3.5 (#884).
Deprecations:
The following functions and geometry attributes and methods will be removed in version 2.0.0.
• ops.cascaded_union• geometry .empty()• geometry .ctypes and .__array_interface__• multi-part geometry .__len__• setting custom attributes on geometry objects
Geometry objects will become immutable in version 2.0.0.
The STRtree class will be entirely changed in 2.0.0. The exact future API is not yet decided, but will be decided before
1.8.0 is released.
Deprecation warnings will be emitted in 1.8a1 when any of these features are used.
The deprecated .to_wkb() and .to_wkt() methods on the geometry objects have been removed.
New features:• Add a normalize() method to geometry classes, exposing the GEOSNormalize algorithm (#1090).
• Initialize STRtree with a capacity of 10 items per node (#1070).
• Load libraries relocated to shapely/.libs by auditwheel versions < 3.1 or relocated to Shapely.libs by auditwheel
versions >= 3.1.
• shapely.ops.voronoi_diagram() computes the Voronoi Diagram of a geometry or geometry collection (#833,#851).
• shapely.validation.make_valid() fixes invalid geometries (#883)
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Bug fixes:• For pyinstaller we now handle the case of more than one GEOS library in the environment, such as when fiona
and rasterio wheels are co-installed with shapely (#1071).
• The ops.split function now splits on touch to eliminate confusing discrepancies between results using multi andsingle part splitters (#1034).
• Several issues with duplication and order of vertices in ops.substring have been fixed (#1008).
Packaging:• The wheels uploaded to PyPI will include GEOS 3.9.1.
1.7.1 (2020-08-20)• STRtree now safely implements the pickle protocol (#915).
• Documentation has been added for minimum_clearance (#875, #874).
• In STRtree.__del__() we guard against calling GEOSSTRtree_destroy when the lgeos module has alreadybeen torn down on exit (#897, #830).
• Documentation for the overlaps() method has been corrected (#920).
• Correct the test in shapely.geometry.base.BaseGeometry.empty() to eliminate memory leaks like theone reported in #745.
• Get free() not from libc but from the processes global symbols (#891), fixing a bug that manifests on OS X 10.15
and 10.16.
• Extracting substrings from complex lines has been made more correct (#848, #849).
• Splitting of complex geometries has been sped up by preparing the input geometry (#871).
• Fix bug in concatenation of function argtypes (#866).
• Improved documentation of STRtree usage (#857).
• Improved handling for empty list or list of lists in GeoJSON coordinates (#852).
• The polylabel algorithm now accounts for polygon holes (#851, #817).
1.7.0 (2020-01-28)
This is the final 1.7.0 release. There have been no changes since 1.7b1.
1.7b1 (2020-01-13)
First beta release.

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1.7a3 (2019-12-31)
New features:• The buffer operation can now be single-sides (#806, #727).
Bug fixes:• Add /usr/local/lib to the list of directories to be searched for the GEOS shared library (#795).
• ops.substring now returns a line with coords in end-to-front order when given a start position that is greater thanthe end position (#628).
• Implement __bool__() for geometry base classes so that bool(geom) returns the logical complement of geom.
is_empty (#754).
• Remove assertion on the number of version-like strings found in the GEOS version string. It could be 2 or 3.
1.7a2 (2019-06-21)• Nearest neighbor search has been added to STRtree (#668).
• Disallow sequences of MultiPolygons as arguments to the MultiPolygon constructor, resolving #588.
• Removed vendorized functools functions previously used to support Python 2.5.
Bug fixes:• Avoid reloading the GEOS shared library when using an installed binary wheel on OS X (#735), resolving issue#553.
• The shapely.ops.orient function can now orient multi polygons and geometry collections as well as polygons(#733).
• Polygons can now be constructed from sequences of point objects as well as sequences of x, y sequences (#732).
• The exterior of an empty polygon is now equal to an empty linear ring (#731).
• The bounds property of an empty point object now returns an empty tuple, consistent with other geometry types(#723).
• Segmentation faults when non-string values are passed to the WKT loader are avoided by #700.
• Failure of ops.substring when the sub linestring coincides with the beginning of the linestring has been fixed(#658).
• Segmentation faults from interpolating on an empty linestring are prevented by #655.
• A missing special case for rectangular polygons has been added to the polylabel algorithm (#644).
• LinearRing can be created from a LineString (#638).
• The prepared geometry validation condition has been tightened in #632 to fix the bug reported in #631.
• Attempting to interpolate an empty geometry no longer results in a segmentation fault, raising ValueError instead(#653).

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1.7a1 (2018-07-29)
New features:• A Python version check is made by the package setup script. Shapely 1.7 supports only Python versions 2.7 and3.4+ (#610).
• Added a new EmptyGeometry class to support GeoPandas (#514).
• Added new shapely.ops.substring function (#459).
• Added new shapely.ops.clip_by_rect function (#583).
• Use DLLs indicated in sys._MEIPASS’ to support PyInstaller frozen apps (#523).
• shapely.wkb.dumps now accepts an srid integer keyword argument to write WKB data including a spatial reference ID in the output data (#593).
Bug fixes:• shapely.geometry.shape can now marshal empty GeoJSON representations (#573).
• An exception is raised when an attempt is made to prepare a PreparedGeometry (#577, #595).
• Keyword arguments have been removed from a geometry object’s wkt property getter (#581, #594).
1.6.4.post1 (2018-01-24)• Fix broken markup in this change log, which restores our nicely formatted readme on PyPI.
1.6.4 (2018-01-24)• Handle a TypeError that can occur when geometries are torn down (#473, #528).
1.6.3 (2017-12-09)• AttributeError is no longer raised when accessing __geo_interface__ of an empty polygon (#450).
• asShape now handles empty coordinates in mappings as shape does (#542). Please note that asShape is likely
to be deprecated in a future version of Shapely.
• Check for length of LineString coordinates in speed mode, preventing crashes when using LineStrings with only
one coordinate (#546).
1.6.2 (2017-10-30)• A 1.6.2.post1 release has been made to fix a problem with macosx wheels uploaded to PyPI.

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1.6.2 (2017-10-26)• Splitting a linestring by one of its end points will now succeed instead of failing with a ValueError (#524,#533).
• Missing documentation of a geometry’s overlaps predicate has been added (#522).
1.6.1 (2017-09-01)• Avoid STRTree crashes due to dangling references (#505) by maintaining references to added geometries.
• Reduce log level to debug when reporting on calls to ctypes CDLL() that don’t succeed and are retried (#515).
• Clarification: applications like GeoPandas that need an empty geometry object should use BaseGeometry()
instead of Point() or Polygon(). An EmptyGeometry class has been added in the master development branch
and will be available in the next non-bugfix release.
1.6.0 (2017-08-21)
Shapely 1.6.0 adds new attributes to existing geometry classes and new functions (split() and polylabel()) to theshapely.ops module. Exceptions are consolidated in a shapely.errors module and logging practices have been improved.
Shapely’s optional features depending on Numpy are now gathered into a requirements set named “vectorized” and these
may be installed like pip install shapely[vectorized].
Much of the work on 1.6.0 was aimed to improve the project’s build and packaging scripts and to minimize run-timedependencies. Shapely now vendorizes packaging to use during builds only and never again invokes the geos-config
utility at run-time.
In addition to the changes listed under the alpha and beta pre-releases below, the following change has been made to
the project:• Project documentation is now hosted at https://shapely.readthedocs.io/en/latest/.
Thank you all for using, promoting, and contributing to the Shapely project.
1.6b5 (2017-08-18)
Bug fixes:• Passing a single coordinate to LineString() with speedups disabled now raises a ValueError as happens withspeedups enabled. This resolves #509.
1.6b4 (2017-02-15)
Bug fixes:• Isolate vendorized packaging in a _vendor directory, remove obsolete dist-info, and remove packaging fromproject requirements (resolves #468).

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1.6b3 (2016-12-31)
Bug fixes:• Level for log messages originating from the GEOS notice handler reduced from WARNING to INFO (#447).
• Permit speedups to be imported again without Numpy (#444).
1.6b2 (2016-12-12)
New features:• Add support for GeometryCollection to shape and asShape functions (#422).
1.6b1 (2016-12-12)
Bug fixes:• Implemented __array_interface__ for empty Points and LineStrings (#403).
1.6a3 (2016-12-01)
Bug fixes:• Remove accidental hard requirement of Numpy (#431).
Packaging:• Put Numpy in an optional requirement set named “vectorized” (#431).
1.6a2 (2016-11-09)
Bug fixes:• Shapely no longer configures logging in geos.py (#415).
Refactoring:• Consolidation of exceptions in shapely.errors.
• UnsupportedGEOSVersionError is raised when GEOS < 3.3.0 (#407).
Packaging:• Added new library search paths to assist Anaconda (#413).
• geos-config will now be bypassed when NO_GEOS_CONFIG env var is set. This allows configuration of Shapely
builds on Linux systems that for whatever reasons do not include the geos-config program (#322).

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1.6a1 (2016-09-14)
New features:• A new error derived from NotImplementedError, with a more useful message, is raised when the GEOS backend
doesn’t support a called method (#216).
• The project() method of LineString has been extended to LinearRing geometries (#286).
• A new minimum_rotated_rectangle attribute has been added to the base geometry class (#354).
• A new shapely.ops.polylabel() function has been added. It computes a point suited for labeling concave
polygons (#395).
• A new shapely.ops.split() function has been added. It splits a geometry by another geometry of lesser
dimension: polygon by line, line by point (#293, #371).
• Polygon.from_bounds() constructs a Polygon from bounding coordinates (#392).
• Support for testing with Numpy 1.4.1 has been added (#301).
• Support creating all kinds of empty geometries from empty lists of Python objects (#397, #404).
Refactoring:• Switch from SingleSidedBuffer() to OffsetCurve() for GEOS >= 3.3 (#270).
• Cython speedups are now enabled by default (#252).
Packaging:• Packaging 16.7, a setup dependency, is vendorized (#314).
• Infrastructure for building manylinux1 wheels has been added (#391).
• The system’s geos-config program is now only checked when setup.py is executed, never during normal useof the module (#244).
• Added new library search paths to assist PyInstaller (#382) and Windows (#343).
1.5.17 (2016-08-31)• Bug fix: eliminate memory leak in geom_factory() (#408).
• Bug fix: remove mention of negative distances in parallel_offset and note that vertices of right hand offset lines
are reversed (#284).
1.5.16 (2016-05-26)• Bug fix: eliminate memory leak when unpickling geometry objects (#384, #385).
• Bug fix: prevent crashes when attempting to pickle a prepared geometry, raising PicklingError instead (#386).
• Packaging: extension modules in the OS X wheels uploaded to PyPI link only libgeos_c.dylib now (you canverify and compare to previous releases with otool -L shapely/vectorized/_vectorized.so).

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1.5.15 (2016-03-29)• Bug fix: use uintptr_t to store pointers instead of long in _geos.pxi, preventing an overflow error (#372, #373).
Note that this bug fix was erroneously reported to have been made in 1.5.14, but was not.
1.5.14 (2016-03-27)• Bug fix: use type() instead of isinstance() when evaluating geometry equality, preventing instances of baseand derived classes from being mistaken for equals (#317).
• Bug fix: ensure that empty geometries are created when constructors have no args (#332, #333).
• Bug fix: support app “freezing” better on Windows by not relying on the __file__ attribute (#342, #377).
• Bug fix: ensure that empty polygons evaluate to be == (#355).
• Bug fix: filter out empty geometries that can cause segfaults when creating and loading STRtrees (#345, #348).
• Bug fix: no longer attempt to reuse GEOS DLLs already loaded by Rasterio or Fiona on OS X (#374, #375).
1.5.13 (2015-10-09)• Restore setup and runtime discovery and loading of GEOS shared library to state at version 1.5.9 (#326).
• On OS X we try to reuse any GEOS shared library that may have been loaded via import of Fiona or Rasterio inorder to avoid a bug involving the GEOS AbstractSTRtree (#324, #327).
1.5.12 (2015-08-27)• Remove configuration of root logger from libgeos.py (#312).
• Skip test_fallbacks on Windows (#308).
• Call setlocale(locale.LC_ALL, “”) instead of resetlocale() on Windows when tearing down the locale test (#308).
• Fix for Sphinx warnings (#309).
• Addition of .cache, .idea, .pyd, .pdb to .gitignore (#310).
1.5.11 (2015-08-23)• Remove packaging module requirement added in 1.5.10 (#305). Distutils can’t parse versions using ‘rc’, but if
we stick to ‘a’ and ‘b’ we will be fine.
1.5.10 (2015-08-22)• Monkey patch affinity module by absolute reference (#299).
• Raise TopologicalError in relate() instead of crashing (#294, #295, #303).

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1.5.9 (2015-05-27)• Fix for 64 bit speedups compatibility (#274).
1.5.8 (2015-04-29)• Setup file encoding bug fix (#254).
• Support for pyinstaller (#261).
• Major prepared geometry operation fix for Windows (#268, #269).
• Major fix for OS X binary wheel (#262).
1.5.7 (2015-03-16)• Test and fix buggy error and notice handlers (#249).
1.5.6 (2015-02-02)• Fix setup regression (#232, #234).
• SVG representation improvements (#233, #237).
1.5.5 (2015-01-20)• MANIFEST changes to restore _geox.pxi (#231).
1.5.4 (2015-01-19)• Fixed OS X binary wheel library load path (#224).
1.5.3 (2015-01-12)• Fixed ownership and potential memory leak in polygonize (#223).
• Wider release of binary wheels for OS X.
1.5.2 (2015-01-04)• Fail installation if GEOS dependency is not met, preventing update breakage (#218, #219).

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1.5.1 (2014-12-04)• Restore geometry hashing (#209).
1.5.0 (2014-12-02)• Affine transformation speedups (#197).
• New == rich comparison (#195).
• Geometry collection constructor (#200).
• ops.snap() backed by GEOSSnap (#201).
• Clearer exceptions in cases of topological invalidity (#203).
1.4.4 (2014-11-02)• Proper conversion of numpy float32 vals to coords (#186).
1.4.3 (2014-10-01)• Fix for endianness bug in WKB writer (#174).
1.4.2 (2014-09-29)• Fix bungled 1.4.1 release (#176).
1.4.1 (2014-09-23)• Return of support for GEOS 3.2 (#176, #178).
1.4.0 (2014-09-08)• SVG representations for IPython’s inline image protocol.
• Efficient and fast vectorized contains().
• Change mitre_limit default to 5.0; raise ValueError with 0.0 (#139).
• Allow mix of tuples and Points in sped-up LineString ctor (#152).
• New STRtree class (#73).
• Add ops.nearest_points() (#147).
• Faster creation of geometric objects from others (cloning) (#165).
• Removal of tests from package.

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1.3.3 (2014-07-23)• Allow single-part geometries as argument to ops.cacaded_union() (#135).
• Support affine transformations of LinearRings (#112).
1.3.2 (2014-05-13)• Let LineString() take a sequence of Points (#130).
1.3.1 (2014-04-22)• More reliable proxy cleanup on exit (#106).
• More robust DLL loading on all platforms (#114).
1.3.0 (2013-12-31)• Include support for Python 3.2 and 3.3 (#56), minimum version is now 2.6.
• Switch to GEOS WKT/WKB Reader/Writer API, with defaults changed to enable 3D output dimensions, and to‘trim’ WKT output for GEOS >=3.3.0.
• Use GEOS version instead of GEOS C API version to determine library capabilities (#65).
1.2.19 (2013-12-30)• Add buffering style options (#55).
1.2.18 (2013-07-23)• Add shapely.ops.transform.
• Permit empty sequences in collection constructors (#49, #50).
• Individual polygons in MultiPolygon.__geo_interface__ are changed to tuples to match Polygon.__geo_interface__ (#51).
• Add shapely.ops.polygonize_full (#57).
1.2.17 (2013-01-27)• Avoid circular import between wkt/wkb and geometry.base by moving calls to GEOS serializers to the latter
module.
• Set _ndim when unpickling (issue #6).
• Don’t install DLLs to Python’s DLL directory (#37).
• Add affinity module of affine transformation (#31).
• Fix NameError that blocked installation with PyPy (#40, #41).

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1.2.16 (2012-09-18)• Add ops.unary_union function.
• Alias ops.cascaded_union to ops.unary_union when GEOS CAPI >= (1,7,0).
• Add geos_version_string attribute to shapely.geos.
• Ensure parent is set when child geometry is accessed.
• Generate _speedups.c using Cython when building from repo when missing, stale, or the build target is “sdist”.
• The is_simple predicate of invalid, self-intersecting linear rings now returns False.
• Remove VERSION.txt from repo, it’s now written by the distutils setup script with value of shapely.__version__.
1.2.15 (2012-06-27)• Eliminate numerical sensitivity in a method chaining test (Debian bug #663210).
• Account for cascaded union of random buffered test points being a polygon or multipolygon (Debian bug#666655).
• Use Cython to build speedups if it is installed.
• Avoid stumbling over SVN revision numbers in GEOS C API version strings.
1.2.14 (2012-01-23)• A geometry’s coords property is now sliceable, yielding a list of coordinate values.
• Homogeneous collections are now sliceable, yielding a new collection of the same type.
1.2.13 (2011-09-16)• Fixed errors in speedups on 32bit systems when GEOS references memory above 2GB.
• Add shapely.__version__ attribute.
• Update the manual.
1.2.12 (2011-08-15)• Build Windows distributions with VC7 or VC9 as appropriate.
• More verbose report on failure to speed up.
• Fix for prepared geometries broken in 1.2.11.
• DO NOT INSTALL 1.2.11

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1.2.11 (2011-08-04)• Ignore AttributeError during exit.
• PyPy 1.5 support.
• Prevent operation on prepared geometry crasher (#12).
• Optional Cython speedups for Windows.
• Linux 3 platform support.
1.2.10 (2011-05-09)• Add optional Cython speedups.
• Add is_cww predicate to LinearRing.
• Add function that forces orientation of Polygons.
• Disable build of speedups on Windows pending packaging work.
1.2.9 (2011-03-31)• Remove extra glob import.
• Move examples to shapely.examples.
• Add box() constructor for rectangular polygons.
• Fix extraneous imports.
1.2.8 (2011-12-03)• New parallel_offset method (#6).
• Support for Python 2.4.
1.2.7 (2010-11-05)• Support for Windows eggs.
1.2.6 (2010-10-21)• The geoms property of an empty collection yields [] instead of a ValueError (#3).
• The coords and geometry type sproperties have the same behavior as above.
• Ensure that z values carry through into products of operations (#4).

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1.2.5 (2010-09-19)• Stop distributing docs/_build.
• Include library fallbacks in test_dlls.py for linux platform.
1.2.4 (2010-09-09)• Raise AttributeError when there’s no backend support for a method.
• Raise OSError if libgeos_c.so (or variants) can’t be found and loaded.
• Add geos_c DLL loading support for linux platforms where find_library doesn’t work.
1.2.3 (2010-08-17)• Add mapping function.
• Fix problem with GEOSisValidReason symbol for GEOS < 3.1.
1.2.2 (2010-07-23)• Add representative_point method.
1.2.1 (2010-06-23)• Fixed bounds of singular polygons.
• Added shapely.validation.explain_validity function (#226).
1.2 (2010-05-27)• Final release.
1.2rc2 (2010-05-26)• Add examples and tests to MANIFEST.in.
• Release candidate 2.
1.2rc1 (2010-05-25)• Release candidate.

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1.2b7 (2010-04-22)• Memory leak associated with new empty geometry state fixed.
1.2b6 (2010-04-13)• Broken GeometryCollection fixed.
1.2b5 (2010-04-09)• Objects can be constructed from others of the same type, thereby making copies. Collections can be constructedfrom sequences of objects, also making copies.
• Collections are now iterators over their component objects.
• New code for manual figures, using the descartes package.
1.2b4 (2010-03-19)• Adds support for the “sunos5” platform.
1.2b3 (2010-02-28)• Only provide simplification implementations for GEOS C API >= 1.5.
1.2b2 (2010-02-19)• Fix cascaded_union bug introduced in 1.2b1 (#212).
1.2b1 (2010-02-18)• Update the README. Remove cruft from setup.py. Add some version 1.2 metadata regarding required Python
version (>=2.5,<3) and external dependency (libgeos_c >= 3.1).
1.2a6 (2010-02-09)• Add accessor for separate arrays of X and Y values (#210).
TODO: fill gap here
1.2a1 (2010-01-20)• Proper prototyping of WKB writer, and avoidance of errors on 64-bit systems (#191).
• Prototype libgeos_c functions in a way that lets py2exe apps import shapely (#189).
1.2 Branched (2009-09-19)

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1.0.12 (2009-04-09)• Fix for references held by topology and predicate descriptors.
1.0.11 (2008-11-20)• Work around bug in GEOS 2.2.3, GEOSCoordSeq_getOrdinate not exported properly (#178).
1.0.10 (2008-11-17)• Fixed compatibility with GEOS 2.2.3 that was broken in 1.0.8 release (#176).
1.0.9 (2008-11-16)• Find and load MacPorts libgeos.
1.0.8 (2008-11-01)• Fill out GEOS function result and argument types to prevent faults on a 64-bit arch.
1.0.7 (2008-08-22)• Polygon rings now have the same dimensions as parent (#168).
• Eliminated reference cycles in polygons (#169).
1.0.6 (2008-07-10)• Fixed adaptation of multi polygon data.
• Raise exceptions earlier from binary predicates.
• Beginning distributing new windows DLLs (#166).
1.0.5 (2008-05-20)• Added access to GEOS polygonizer function.
• Raise exception when insufficient coordinate tuples are passed to LinearRing constructor (#164).
1.0.4 (2008-05-01)• Disentangle Python and topological equality (#163).
• Add shape(), a factory that copies coordinates from a geo interface provider. To be used instead of asShape()
unless you really need to store coordinates outside shapely for efficient use in other code.
• Cache GEOS geometries in adapters (#163).

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1.0.3 (2008-04-09)• Do not release GIL when calling GEOS functions (#158).
• Prevent faults when chaining multiple GEOS operators (#159).
1.0.2 (2008-02-26)• Fix loss of dimensionality in polygon rings (#155).
1.0.1 (2008-02-08)• Allow chaining expressions involving coordinate sequences and geometry parts (#151).
• Protect against abnormal use of coordinate accessors (#152).
• Coordinate sequences now implement the numpy array protocol (#153).
1.0 (2008-01-18)• Final release.
1.0 RC2 (2008-01-16)• Added temporary solution for #149.
1.0 RC1 (2008-01-14)• First release candidate

1.1.9 Frequently asked questions and answers
I installed shapely in a conda environment using pip. Why doesn’t it work?
Shapely versions < 2.0 load a GEOS shared library using ctypes. It’s not uncommon for users to have multiple copies
of GEOS libs on their system. Loading the correct one is complicated and shapely has a number of platform-dependent
GEOS library loading bugs. The project has particularly poor support for finding the correct GEOS library for ashapely package installed from PyPI into a conda environment. We recommend that conda users always get shapelyfrom conda-forge.
Are there references for the algorithms used by shapely?
Generally speaking, shapely’s predicates and operations are derived from methods of the same name from GEOS and
the JTS Topology Suite. See the JTS FAQ for references describing the JTS algorithms.

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I used .buffer() on a geometry with Z coordinates. Where did the Z coordinates go?
The buffer algorithm in GEOS is purely two-dimensional and discards any Z coordinates. This is generally the case for
the GEOS algorithms.

1.2 The Shapely User Manual
Author Sean Gillies, <sean.gillies@gmail.com>
Version 1.8.0
Date Feb 16, 2022
Copyright This work is licensed under a Creative Commons Attribution 3.0 United States License.
Abstract This document explains how to use the Shapely Python package for computational geometry.

1.2.1 Introduction
Deterministic spatial analysis is an important component of computational approaches to problems in agriculture,
ecology, epidemiology, sociology, and many other fields. What is the surveyed perimeter/area ratio of these patches ofanimal habitat? Which properties in this town intersect with the 50-year flood contour from this new flooding model?
What are the extents of findspots for ancient ceramic wares with maker’s marks “A” and “B”, and where do the extents
overlap? What’s the path from home to office that best skirts identified zones of location based spam? These are justa few of the possible questions addressable using non-statistical spatial analysis, and more specifically, computational
geometry.
Shapely is a Python package for set-theoretic analysis and manipulation of planar features using (via Python’s ctypes
module) functions from the well known and widely deployed GEOS library. GEOS, a port of the Java Topology Suite(JTS), is the geometry engine of the PostGIS spatial extension for the PostgreSQL RDBMS. The designs of JTS and
GEOS are largely guided by the Open Geospatial Consortium’s Simple Features Access Specification1 and Shapely
adheres mainly to the same set of standard classes and operations. Shapely is thereby deeply rooted in the conventionsof the geographic information systems (GIS) world, but aspires to be equally useful to programmers working on nonconventional problems.
The first premise of Shapely is that Python programmers should be able to perform PostGIS type geometry operations
outside of an RDBMS. Not all geographic data originate or reside in a RDBMS or are best processed using SQL.
We can load data into a spatial RDBMS to do work, but if there’s no mandate to manage (the “M” in “RDBMS”)
the data over time in the database we’re using the wrong tool for the job. The second premise is that the persistence,
serialization, and map projection of features are significant, but orthogonal problems. You may not need a hundred
GIS format readers and writers or the multitude of State Plane projections, and Shapely doesn’t burden you with them.
The third premise is that Python idioms trump GIS (or Java, in this case, since the GEOS library is derived from JTS,a Java project) idioms.
If you enjoy and profit from idiomatic Python, appreciate packages that do one thing well, and agree that a spatiallyenabled RDBMS is often enough the wrong tool for your computational geometry job, Shapely might be for you.
1 John R. Herring, Ed., “OpenGIS Implementation Specification for Geographic information - Simple feature access - Part 1: Common architecture,” Oct. 2006.

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Spatial Data Model
The fundamental types of geometric objects implemented by Shapely are points, curves, and surfaces. Each is associated with three sets of (possibly infinite) points in the plane. The interior, boundary, and exterior sets of a feature are
mutually exclusive and their union coincides with the entire plane2 .
• A Point has an interior set of exactly one point, a boundary set of exactly no points, and an exterior set of allother points. A Point has a topological dimension of 0.
• A Curve has an interior set consisting of the infinitely many points along its length (imagine a Point dragged inspace), a boundary set consisting of its two end points, and an exterior set of all other points. A Curve has a
topological dimension of 1.
• A Surface has an interior set consisting of the infinitely many points within (imagine a Curve dragged in space tocover an area), a boundary set consisting of one or more Curves, and an exterior set of all other points including
those within holes that might exist in the surface. A Surface has a topological dimension of 2.
That may seem a bit esoteric, but will help clarify the meanings of Shapely’s spatial predicates, and it’s as deep into
theory as this manual will go. Consequences of point-set theory, including some that manifest themselves as “gotchas”,
for different classes will be discussed later in this manual.
The point type is implemented by a Point class; curve by the LineString and LinearRing classes; and surface by a Polygon class. Shapely implements no smooth (i.e. having continuous tangents) curves. All curves must be approximatedby linear splines. All rounded patches must be approximated by regions bounded by linear splines.
Collections of points are implemented by a MultiPoint class, collections of curves by a MultiLineString class, andcollections of surfaces by a MultiPolygon class. These collections aren’t computationally significant, but are usefulfor modeling certain kinds of features. A Y-shaped line feature, for example, is well modeled as a whole by a MultiLineString.
The standard data model has additional constraints specific to certain types of geometric objects that will be discussed
in following sections of this manual.
See also https://web.archive.org/web/20160719195511/http://www.vividsolutions.com/jts/discussion.htm for more illustrations of this data model.
Relationships
The spatial data model is accompanied by a group of natural language relationships between geometric objects – contains, intersects, overlaps, touches, etc. – and a theoretical framework for understanding them using the 3x3 matrix ofthe mutual intersections of their component point sets3 : the DE-9IM. A comprehensive review of the relationships interms of the DE-9IM is found in4 and will not be reiterated in this manual.
2 M.J. Egenhofer and John R. Herring, Categorizing Binary Topological Relations Between Regions, Lines, and Points in Geographic Databases,
Orono, ME: University of Maine, 1991.
3 E. Clementini, P. Di Felice, and P. van Oosterom, “A Small Set of Formal Topological Relationships Suitable for End-User Interaction,” Third
International Symposium on Large Spatial Databases (SSD). Lecture Notes in Computer Science no. 692, David Abel and Beng Chin Ooi, Eds.,
Singapore: Springer Verlag, 1993, pp. 277-295.
4 C. Strobl, “Dimensionally Extended Nine-Intersection Model (DE-9IM),” Encyclopedia of GIS, S. Shekhar and H. Xiong, Eds., Springer, 2008,
pp. 240-245. [PDF]1.2. The Shapely User Manual27

Shapely Documentation, Release 1.8.1

Operations
Following the JTS technical specs5 , this manual will make a distinction between constructive (buffer, convex hull) andset-theoretic operations (intersection, union, etc.). The individual operations will be fully described in a following
section of the manual.
Coordinate Systems
Even though the Earth is not flat – and for that matter not exactly spherical – there are many analytic problems thatcan be approached by transforming Earth features to a Cartesian plane, applying tried and true algorithms, and thentransforming the results back to geographic coordinates. This practice is as old as the tradition of accurate paper maps.
Shapely does not support coordinate system transformations. All operations on two or more features presume that thefeatures exist in the same Cartesian plane.

1.2.2 Geometric Objects
Geometric objects are created in the typical Python fashion, using the classes themselves as instance factories. A
few of their intrinsic properties will be discussed in this sections, others in the following sections on operations andserializations.
Instances of Point, LineString, and LinearRing have as their most important attribute a finite sequence of coordinates that determines their interior, boundary, and exterior point sets. A line string can be determined by as few as
2 points, but contains an infinite number of points. Coordinate sequences are immutable. A third z coordinate value
may be used when constructing instances, but has no effect on geometric analysis. All operations are performed in thex-y plane.
In all constructors, numeric values are converted to type float. In other words, Point(0, 0) and Point(0.0, 0.0)
produce geometrically equivalent instances. Shapely does not check the topological simplicity or validity of instances
when they are constructed as the cost is unwarranted in most cases. Validating factories are easily implemented using
the :attr:is_valid predicate by users that require them.
Note: Shapely is a planar geometry library and z, the height above or below the plane, is ignored in geometric analysis.
There is a potential pitfall for users here: coordinate tuples that differ only in z are not distinguished from each otherand their application can result in surprisingly invalid geometry objects. For example, LineString([(0, 0, 0),(0, 0, 1)]) does not return a vertical line of unit length, but an invalid line in the plane with zero length. Similarly,
Polygon([(0, 0, 0), (0, 0, 1), (1, 1, 1)]) is not bounded by a closed ring and is invalid.

General Attributes and Methods
object.area
Returns the area (float) of the object.
object.bounds
Returns a (minx, miny, maxx, maxy) tuple (float values) that bounds the object.
object.length
Returns the length (float) of the object.
object.minimum_clearance
Returns the smallest distance by which a node could be moved to produce an invalid geometry.
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This can be thought of as a measure of the robustness of a geometry, where larger values of minimum clearance
indicate a more robust geometry. If no minimum clearance exists for a geometry, such as a point, this will returnmath.infinity.
New in Shapely 1.7.1
Requires GEOS 3.6 or higher.
>>> from shapely.geometry import Polygon>>> Polygon([[0, 0], [1, 0], [1, 1], [0, 1], [0, 0]]).minimum_clearance
1.0
object.geom_type
Returns a string specifying the Geometry Type of the object in accordance with? .
>>> Point(0, 0).geom_type'Point'object.distance(other)
Returns the minimum distance (float) to the other geometric object.
>>> Point(0,0).distance(Point(1,1))
1.4142135623730951
object.hausdorff_distance(other)
Returns the Hausdorff distance (float) to the other geometric object. The Hausdorff distance between two
geometries is the furthest distance that a point on either geometry can be from the nearest point to it on the othergeometry.
New in Shapely 1.6.0>>> point = Point(1, 1)>>> line = LineString([(2, 0), (2, 4), (3, 4)])>>> point.hausdorff_distance(line)
3.605551275463989>>> point.distance(Point(3, 4))3.605551275463989
object.representative_point()
Returns a cheaply computed point that is guaranteed to be within the geometric object.
Note: This is not in general the same as the centroid.
>>> donut = Point(0, 0).buffer(2.0).difference(Point(0, 0).buffer(1.0))>>> donut.centroid.wkt'POINT (-0.0000000000000001 -0.0000000000000000)'>>> donut.representative_point().wkt'POINT (-1.5000000000000000 0.0000000000000000)'

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Pointsclass Point(coordinates)
The Point constructor takes positional coordinate values or point tuple parameters.
>>> from shapely.geometry import Point>>> point = Point(0.0, 0.0)>>> q = Point((0.0, 0.0))
A Point has zero area and zero length.
>>> point.area
0.0>>> point.length
0.0
Its x-y bounding box is a (minx, miny, maxx, maxy) tuple.
>>> point.bounds(0.0, 0.0, 0.0, 0.0)
Coordinate values are accessed via coords, x, y, and z properties.
>>> list(point.coords)[(0.0, 0.0)]>>> point.x
0.0>>> point.y
0.0
Coordinates may also be sliced. New in version 1.2.14.
>>> point.coords[:][(0.0, 0.0)]
The Point constructor also accepts another Point instance, thereby making a copy.
>>> Point(point)<shapely.geometry.point.Point object at 0x...>LineStrings
class LineString(coordinates)
The LineString constructor takes an ordered sequence of 2 or more (x, y[, z]) point tuples.
The constructed LineString object represents one or more connected linear splines between the points. Repeated points
in the ordered sequence are allowed, but may incur performance penalties and should be avoided. A LineString may
cross itself (i.e. be complex and not simple).
Figure 1. A simple LineString on the left, a complex LineString on the right. The (MultiPoint) boundary of each isshown in black, the other points that describe the lines are shown in grey.
A LineString has zero area and non-zero length.

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Shapely Documentation, Release 1.8.1a) simple

3
2

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0

1

1

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2

b) complex

3

3

4

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>>> from shapely.geometry import LineString>>> line = LineString([(0, 0), (1, 1)])>>> line.area
0.0>>> line.length
1.4142135623730951
Its x-y bounding box is a (minx, miny, maxx, maxy) tuple.
>>> line.bounds(0.0, 0.0, 1.0, 1.0)
The defining coordinate values are accessed via the coords property.
>>> len(line.coords)
2>>> list(line.coords)[(0.0, 0.0), (1.0, 1.0)]
Coordinates may also be sliced. New in version 1.2.14.
>>> point.coords[:][(0.0, 0.0), (1.0, 1.0)]>>> point.coords[1:][(1.0, 1.0)]
The constructor also accepts another LineString instance, thereby making a copy.

1.2. The Shapely User Manual31Shapely Documentation, Release 1.8.1>>> LineString(line)<shapely.geometry.linestring.LineString object at 0x...>A LineString may also be constructed using a sequence of mixed Point instances or coordinate tuples. The individual
coordinates are copied into the new object.
>>> LineString([Point(0.0, 1.0), (2.0, 3.0), Point(4.0, 5.0)])<shapely.geometry.linestring.LineString object at 0x...>LinearRings
class LinearRing(coordinates)
The LinearRing constructor takes an ordered sequence of (x, y[, z]) point tuples.
The sequence may be explicitly closed by passing identical values in the first and last indices. Otherwise, the sequence
will be implicitly closed by copying the first tuple to the last index. As with a LineString, repeated points in the orderedsequence are allowed, but may incur performance penalties and should be avoided. A LinearRing may not cross itself,
and may not touch itself at a single point.

a) valid

3
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2

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0

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b) invalid

3

2

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Figure 2. A valid LinearRing on the left, an invalid self-touching LinearRing on the right. The points that describe therings are shown in grey. A ring’s boundary is empty.
Note: Shapely will not prevent the creation of such rings, but exceptions will be raised when they are operated on.
A LinearRing has zero area and non-zero length.

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Shapely Documentation, Release 1.8.1>>> from shapely.geometry.polygon import LinearRing>>> ring = LinearRing([(0, 0), (1, 1), (1, 0)])>>> ring.area
0.0>>> ring.length
3.4142135623730949
Its x-y bounding box is a (minx, miny, maxx, maxy) tuple.
>>> ring.bounds(0.0, 0.0, 1.0, 1.0)
Defining coordinate values are accessed via the coords property.
>>> len(ring.coords)
4>>> list(ring.coords)[(0.0, 0.0), (1.0, 1.0), (1.0, 0.0), (0.0, 0.0)]
The LinearRing constructor also accepts another LineString or LinearRing instance, thereby making a copy.
>>> LinearRing(ring)<shapely.geometry.polygon.LinearRing object at 0x...>As with LineString, a sequence of Point instances is not a valid constructor parameter.
Polygons
class Polygon(shell[, holes=None ])The Polygon constructor takes two positional parameters. The first is an ordered sequence of (x, y[, z])point tuples and is treated exactly as in the LinearRing case. The second is an optional unordered sequence of
ring-like sequences specifying the interior boundaries or “holes” of the feature.
Rings of a valid Polygon may not cross each other, but may touch at a single point only. Again, Shapely will not preventthe creation of invalid features, but exceptions will be raised when they are operated on.
Figure 3. On the left, a valid Polygon with one interior ring that touches the exterior ring at one point, and on theright a Polygon that is invalid because its interior ring touches the exterior ring at more than one point. The points that
describe the rings are shown in grey.
Figure 4. On the left, a Polygon that is invalid because its exterior and interior rings touch along a line, and on theright, a Polygon that is invalid because its interior rings touch along a line.
A Polygon has non-zero area and non-zero length.
>>> from shapely.geometry import Polygon>>> polygon = Polygon([(0, 0), (1, 1), (1, 0)])>>> polygon.area
0.5>>> polygon.length
3.4142135623730949
Its x-y bounding box is a (minx, miny, maxx, maxy) tuple.

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Shapely Documentation, Release 1.8.1>>> polygon.bounds(0.0, 0.0, 1.0, 1.0)
Component rings are accessed via exterior and interiors properties.
>>> list(polygon.exterior.coords)[(0.0, 0.0), (1.0, 1.0), (1.0, 0.0), (0.0, 0.0)]>>> list(polygon.interiors)[]
The Polygon constructor also accepts instances of LineString and LinearRing.
>>> coords = [(0, 0), (1, 1), (1, 0)]>>> r = LinearRing(coords)>>> s = Polygon(r)>>> s.area
0.5>>> t = Polygon(s.buffer(1.0).exterior, [r])>>> t.area
6.5507620529190334
Rectangular polygons occur commonly, and can be conveniently constructed using the shapely.geometry.box()
function.
shapely.geometry.box(minx, miny, maxx, maxy, ccw=True)
Makes a rectangular polygon from the provided bounding box values, with counter-clockwise order by default.
New in version 1.2.9.
For example:>>> from shapely.geometry import box>>> b = box(0.0, 0.0, 1.0, 1.0)>>> b<shapely.geometry.polygon.Polygon object at 0x...>>>> list(b.exterior.coords)[(1.0, 0.0), (1.0, 1.0), (0.0, 1.0), (0.0, 0.0), (1.0, 0.0)]
This is the first appearance of an explicit polygon handedness in Shapely.
To obtain a polygon with a known orientation, use shapely.geometry.polygon.orient():
shapely.geometry.polygon.orient(polygon, sign=1.0)
Returns a properly oriented copy of the given polygon. The signed area of the result will have the given sign. A
sign of 1.0 means that the coordinates of the product’s exterior ring will be oriented counter-clockwise and theinterior rings (holes) will be oriented clockwise.
New in version 1.2.10.

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Collections
Heterogeneous collections of geometric objects may result from some Shapely operations. For example, two
LineStrings may intersect along a line and at a point. To represent these kind of results, Shapely provides frozenset-like,
immutable collections of geometric objects. The collections may be homogeneous (MultiPoint etc.) or heterogeneous.
>>> a = LineString([(0, 0), (1, 1), (1,2), (2,2)])>>> b = LineString([(0, 0), (1, 1), (2,1), (2,2)])>>> x = a.intersection(b)>>> x<shapely.geometry.collection.GeometryCollection object at 0x...>>>> from pprint import pprint>>> pprint(list(x))[<shapely.geometry.point.Point object at 0x...>,<shapely.geometry.linestring.LineString object at 0x...>]

a) lines

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Figure 5. a) a green and a yellow line that intersect along a line and at a single point; b) the intersection (in blue) is acollection containing one LineString and one Point.
Members of a GeometryCollection are accessed via the geoms property or via the iterator protocol using in or list().
>>> pprint(list(x.geoms))[<shapely.geometry.point.Point object at 0x...>,<shapely.geometry.linestring.LineString object at 0x...>]>>> pprint(list(x))[<shapely.geometry.point.Point object at 0x...>,<shapely.geometry.linestring.LineString object at 0x...>]
Collections can also be sliced.

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Shapely Documentation, Release 1.8.1>>> from shapely.geometry import MultiPoint>>> m = MultiPoint([(0, 0), (1, 1), (1,2), (2,2)])>>> m[:1].wkt'MULTIPOINT (0.0000000000000000 0.0000000000000000)'>>> m[3:].wkt'MULTIPOINT (2.0000000000000000 2.0000000000000000)'>>> m[4:].wkt'GEOMETRYCOLLECTION EMPTY'
New in version 1.2.14.
Note: When possible, it is better to use one of the homogeneous collection types described below.

Collections of Points
class MultiPoint(points)
The MultiPoint constructor takes a sequence of (x, y[, z ]) point tuples.
A MultiPoint has zero area and zero length.
>>> from shapely.geometry import MultiPoint>>> points = MultiPoint([(0.0, 0.0), (1.0, 1.0)])>>> points.area
0.0>>> points.length
0.0
Its x-y bounding box is a (minx, miny, maxx, maxy) tuple.
>>> points.bounds(0.0, 0.0, 1.0, 1.0)
Members of a multi-point collection are accessed via the geoms property or via the iterator protocol using in or list().
>>> import pprint>>> pprint.pprint(list(points.geoms))[<shapely.geometry.point.Point object at 0x...>,<shapely.geometry.point.Point object at 0x...>]>>> pprint.pprint(list(points))[<shapely.geometry.point.Point object at 0x...>,<shapely.geometry.point.Point object at 0x...>]
The constructor also accepts another MultiPoint instance or an unordered sequence of Point instances, thereby making
copies.
>>> MultiPoint([Point(0, 0), Point(1, 1)])
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Collections of Linesclass MultiLineString(lines)
The MultiLineString constructor takes a sequence of line-like sequences or objects.

a) simple

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Figure 6. On the left, a simple, disconnected MultiLineString, and on the right, a non-simple MultiLineString. The
points defining the objects are shown in gray, the boundaries of the objects in black.
A MultiLineString has zero area and non-zero length.
>>> from shapely.geometry import MultiLineString>>> coords = [((0, 0), (1, 1)), ((-1, 0), (1, 0))]>>> lines = MultiLineString(coords)>>> lines.area
0.0>>> lines.length
3.4142135623730949
Its x-y bounding box is a (minx, miny, maxx, maxy) tuple.
>>> lines.bounds(-1.0, 0.0, 1.0, 1.0)
Its members are instances of LineString and are accessed via the geoms property or via the iterator protocol using inor list().
>>> len(lines.geoms)2>>> pprint.pprint(list(lines.geoms))(continues on next page)

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Shapely Documentation, Release 1.8.1(continued from previous page)[<shapely.geometry.linestring.LineString object at 0x...>,<shapely.geometry.linestring.LineString object at 0x...>]>>> pprint.pprint(list(lines))[<shapely.geometry.linestring.LineString object at 0x...>,<shapely.geometry.linestring.LineString object at 0x...>]
The constructor also accepts another instance of MultiLineString or an unordered sequence of LineString instances,
thereby making copies.
>>> MultiLineString(lines)<shapely.geometry.multilinestring.MultiLineString object at 0x...>>>> MultiLineString(lines.geoms)<shapely.geometry.multilinestring.MultiLineString object at 0x...>

Collections of Polygons
class MultiPolygon(polygons)
The MultiPolygon constructor takes a sequence of exterior ring and hole list tuples: [((a1, . . . , aM), [(b1, . . . ,bN), . . . ]), . . . ].
More clearly, the constructor also accepts an unordered sequence of Polygon instances, thereby making copies.
>>> polygons = MultiPolygon([polygon, s, t])>>> len(polygons.geoms)
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Figure 7. On the left, a valid MultiPolygon with 2 members, and on the right, a MultiPolygon that is invalid because its
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members touch at an infinite number of points (along a line).
Its x-y bounding box is a (minx, miny, maxx, maxy) tuple.
>>> polygons.bounds(-1.0, -1.0, 2.0, 2.0)
Its members are instances of Polygon and are accessed via the geoms property.
>>> len(polygons.geoms)
3

Empty features
An “empty” feature is one with a point set that coincides with the empty set; not None, but like set([]). Empty
features can be created by calling the various constructors with no arguments. Almost no operations are supported byempty features.
>>> line = LineString()>>> line.is_empty
True>>> line.length
0.0>>> line.bounds()>>> line.coords[]
The coordinates of a empty feature can be set, after which the geometry is no longer empty.
>>> line.coords = [(0, 0), (1, 1)]>>> line.is_empty
False>>> line.length
1.4142135623730951>>> line.bounds(0.0, 0.0, 1.0, 1.0)

Coordinate sequences
The list of coordinates that describe a geometry are represented as the CoordinateSequence object. These sequences
should not be initialised directly, but can be accessed from an existing geometry as the Geometry.coords property.
>>> line = LineString([(0, 1), (2, 3), (4, 5)])>>> line.coords<shapely.coords.CoordinateSequence object at 0x00000276EED1C7F0>
Coordinate sequences can be indexed, sliced and iterated over as if they were a list of coordinate tuples.
>>> line.coords[0](0.0, 1.0)>>> line.coords[1:](continues on next page)

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Shapely Documentation, Release 1.8.1(continued from previous page)[(2.0, 3.0), (4.0, 5.0)]>>> for x, y in line.coords:...
print("x={}, y={}".format(x, y))...
x=0.0, y=1.0
x=2.0, y=3.0
x=4.0, y=5.0
Polygons have a coordinate sequence for their exterior and each of their interior rings.
>>> poly = Polygon([(0, 0), (0, 1), (1, 1), (0, 0)])>>> poly.exterior.coords<shapely.coords.CoordinateSequence object at 0x00000276EED1C048>
Multipart geometries do not have a coordinate sequence. Instead the coordinate sequences are stored on their component
geometries.
>>> p = MultiPoint([(0, 0), (1, 1), (2, 2)])>>> p[2].coords<shapely.coords.CoordinateSequence object at 0x00000276EFB9B320>

Linear Referencing Methods
It can be useful to specify position along linear features such as LineStrings and MultiLineStrings with a 1-dimensionalreferencing system. Shapely supports linear referencing based on length or distance, evaluating the distance along ageometric object to the projection of a given point, or the point at a given distance along the object.
object.interpolate(distance[, normalized=False ])Return a point at the specified distance along a linear geometric object.
If the normalized arg is True, the distance will be interpreted as a fraction of the geometric object’s length.
>>> ip = LineString([(0, 0), (0, 1), (1, 1)]).interpolate(1.5)>>> ip<shapely.geometry.point.Point object at 0x740570>>>> ip.wkt'POINT (0.5000000000000000 1.0000000000000000)'>>> LineString([(0, 0), (0, 1), (1, 1)]).interpolate(0.75, normalized=True).wkt'POINT (0.5000000000000000 1.0000000000000000)'
object.project(other [, normalized=False ])Returns the distance along this geometric object to a point nearest the other object.
If the normalized arg is True, return the distance normalized to the length of the object. The project() method is
the inverse of interpolate().
>>> LineString([(0, 0), (0, 1), (1, 1)]).project(ip)1.5>>> LineString([(0, 0), (0, 1), (1, 1)]).project(ip, normalized=True)
0.75
For example, the linear referencing methods might be used to cut lines at a specified distance.

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def cut(line, distance):# Cuts a line in two at a distance from its starting point
if distance <= 0.0 or distance >= line.length:
return [LineString(line)]coords = list(line.coords)
for i, p in enumerate(coords):
pd = line.project(Point(p))
if pd == distance:
return [LineString(coords[:i+1]),
LineString(coords[i:])]
if pd > distance:
cp = line.interpolate(distance)
return [LineString(coords[:i] + [(cp.x, cp.y)]),LineString([(cp.x, cp.y)] + coords[i:])]>>> line = LineString([(0, 0), (1, 0), (2, 0), (3, 0), (4, 0), (5, 0)])>>> pprint([list(x.coords) for x in cut(line, 1.0)])[[(0.0, 0.0), (1.0, 0.0)],[(1.0, 0.0), (2.0, 0.0), (3.0, 0.0), (4.0, 0.0), (5.0, 0.0)]]>>> pprint([list(x.coords) for x in cut(line, 2.5)])[[(0.0, 0.0), (1.0, 0.0), (2.0, 0.0), (2.5, 0.0)],[(2.5, 0.0), (3.0, 0.0), (4.0, 0.0), (5.0, 0.0)]]

1.2.3 Predicates and Relationships
Objects of the types explained in Geometric Objects provide standard? predicates as attributes (for unary predicates)
and methods (for binary predicates). Whether unary or binary, all return True or False.
Unary PredicatesStandard unary predicates are implemented as read-only property attributes. An example will be shown for each.
object.has_z
Returns True if the feature has not only x and y, but also z coordinates for 3D (or so-called, 2.5D) geometries.
>>> Point(0, 0).has_z
False>>> Point(0, 0, 0).has_z
Trueobject.is_ccw
Returns True if coordinates are in counter-clockwise order (bounding a region with positive signed area). Thismethod applies to LinearRing objects only.
New in version 1.2.10.
>>> LinearRing([(1,0), (1,1), (0,0)]).is_ccw
True
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Shapely Documentation, Release 1.8.1>>> ring = LinearRing([(0,0), (1,1), (1,0)])>>> ring.is_ccw
False>>> ring.coords = list(ring.coords)[::-1]>>> ring.is_ccw
Trueobject.is_empty
Returns True if the feature’s interior and boundary (in point set terms) coincide with the empty set.
>>> Point().is_empty
True>>> Point(0, 0).is_empty
False

Note: With the help of the operator module’s attrgetter() function, unary predicates such as is_empty can beeasily used as predicates for the built in filter() or itertools.ifilter().
>>> from operator import attrgetter>>> empties = filter(attrgetter('is_empty'), [Point(), Point(0, 0)])>>> len(empties)
1
object.is_ring
Returns True if the feature is a closed and simple LineString. A closed feature’s boundary coincides with theempty set.
>>> LineString([(0, 0), (1, 1), (1, -1)]).is_ring
False>>> LinearRing([(0, 0), (1, 1), (1, -1)]).is_ring
TrueThis property is applicable to LineString and LinearRing instances, but meaningless for others.
object.is_simple
Returns True if the feature does not cross itself.
Note: The simplicity test is meaningful only for LineStrings and LinearRings.
>>> LineString([(0, 0), (1, 1), (1, -1), (0, 1)]).is_simple
False
Operations on non-simple LineStrings are fully supported by Shapely.
object.is_valid
Returns True if a feature is “valid” in the sense of? .
Note: The validity test is meaningful only for Polygons and MultiPolygons. True is always returned for other typesof geometries.

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A valid Polygon may not possess any overlapping exterior or interior rings. A valid MultiPolygon may not collect any
overlapping polygons. Operations on invalid features may fail.
>>> MultiPolygon([Point(0, 0).buffer(2.0), Point(1, 1).buffer(2.0)]).is_valid
False
The two points above are close enough that the polygons resulting from the buffer operations (explained in a following
section) overlap.
Note: The is_valid predicate can be used to write a validating decorator that could ensure that only valid objects
are returned from a constructor function.
from functools import wraps
def validate(func):@wraps(func)
def wrapper(*args, **kwargs):
ob = func(*args, **kwargs)
if not ob.is_valid:raise TopologicalError("Given arguments do not determine a valid geometric object")return ob
return wrapper>>> @validate... def ring(coordinates):...
return LinearRing(coordinates)...
>>> coords = [(0, 0), (1, 1), (1, -1), (0, 1)]>>> ring(coords)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "<stdin>", line 7, in wrapper
shapely.geos.TopologicalError: Given arguments do not determine a valid geometric object

Binary PredicatesStandard binary predicates are implemented as methods. These predicates evaluate topological, set-theoretic relationships. In a few cases the results may not be what one might expect starting from different assumptions. All take anothergeometric object as argument and return True or False.
object.__eq__(other)
Returns True if the two objects are of the same geometric type, and the coordinates of the two objects match
precisely.
object.equals(other)
Returns True if the set-theoretic boundary, interior, and exterior of the object coincide with those of the other.
The coordinates passed to the object constructors are of these sets, and determine them, but are not the entirety of thesets. This is a potential “gotcha” for new users. Equivalent lines, for example, can be constructed differently.
>>> a = LineString([(0, 0), (1, 1)])>>> b = LineString([(0, 0), (0.5, 0.5), (1, 1)])
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Shapely Documentation, Release 1.8.1(continued from previous page)>>> c = LineString([(0, 0), (0, 0), (1, 1)])>>> a.equals(b)True>>> a == b
False>>> b.equals(c)True>>> b == cFalse
object.almost_equals(other [, decimal=6 ])Returns True if the object is approximately equal to the other at all points to specified decimal place precision.
object.contains(other)
Returns True if no points of other lie in the exterior of the object and at least one point of the interior of otherlies in the interior of object.
This predicate applies to all types, and is inverse to within(). The expression a.contains(b) == b.within(a)always evaluates to True.
>>> coords = [(0, 0), (1, 1)]>>> LineString(coords).contains(Point(0.5, 0.5))
True>>> Point(0.5, 0.5).within(LineString(coords))
True
A line’s endpoints are part of its boundary and are therefore not contained.
>>> LineString(coords).contains(Point(1.0, 1.0))
False

Note: Binary predicates can be used directly as predicates for filter() or itertools.ifilter().
>>> line = LineString(coords)>>> contained = filter(line.contains, [Point(), Point(0.5, 0.5)])>>> len(contained)1>>> [p.wkt for p in contained]['POINT (0.5000000000000000 0.5000000000000000)']
object.covers(other)
Returns True if every point of other is a point on the interior or boundary of object. This is similar to object.
contains(other) except that this does not require any interior points of other to lie in the interior of object.
object.covered_by(other)
Returns True if every point of object is a point on the interior or boundary of other. This is equivalent toother.covers(object).
New in version 1.8.
object.crosses(other)
Returns True if the interior of the object intersects the interior of the other but does not contain it, and thedimension of the intersection is less than the dimension of the one or the other.

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Shapely Documentation, Release 1.8.1>>> LineString(coords).crosses(LineString([(0, 1), (1, 0)]))True
A line does not cross a point that it contains.
>>> LineString(coords).crosses(Point(0.5, 0.5))
False
object.disjoint(other)
Returns True if the boundary and interior of the object do not intersect at all with those of the other.
>>> Point(0, 0).disjoint(Point(1, 1))
True
This predicate applies to all types and is the inverse of intersects().
object.intersects(other)
Returns True if the boundary or interior of the object intersect in any way with those of the other.
In other words, geometric objects intersect if they have any boundary or interior point in common.
object.overlaps(other)
Returns True if the geometries have more than one but not all points in common, have the same dimension, andthe intersection of the interiors of the geometries has the same dimension as the geometries themselves.
object.touches(other)
Returns True if the objects have at least one point in common and their interiors do not intersect with any partof the other.
Overlapping features do not therefore touch, another potential “gotcha”. For example, the following lines touch at (1,1), but do not overlap.
>>> a = LineString([(0, 0), (1, 1)])>>> b = LineString([(1, 1), (2, 2)])>>> a.touches(b)Trueobject.within(other)
Returns True if the object’s boundary and interior intersect only with the interior of the other (not its boundaryor exterior).
This applies to all types and is the inverse of contains().
Used in a sorted() key, within() makes it easy to spatially sort objects. Let’s say we have 4 stereotypic features: apoint that is contained by a polygon which is itself contained by another polygon, and a free spirited point containedby none>>> a = Point(2, 2)>>> b = Polygon([[1, 1], [1, 3], [3, 3], [3, 1]])>>> c = Polygon([[0, 0], [0, 4], [4, 4], [4, 0]])>>> d = Point(-1, -1)
and that copies of these are collected into a list>>> features = [c, a, d, b, c]

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that we’d prefer to have ordered as [d, c, c, b, a] in reverse containment order. As explained in the Python Sorting
HowTo, we can define a key function that operates on each list element and returns a value for comparison. Our key
function will be a wrapper class that implements __lt__() using Shapely’s binary within() predicate.
class Within:def __init__(self, o):self.o = o
def __lt__(self, other):return self.o.within(other.o)As the howto says, the less than comparison is guaranteed to be used in sorting. That’s what we’ll rely on to spatiallysort, and the reason why we use within() in reverse instead of contains(). Trying it out on features d and c, we
see that it works.
>>> d < cTrue>>> Within(d) < Within(c)False
It also works on the list of features, producing the order we want.
>>> [d, c, c, b, a] == sorted(features, key=Within, reverse=True)
True

DE-9IM Relationships
The relate() method tests all the DE-9IM? relationships between objects, of which the named relationship predicates
above are a subset.
object.relate(other)
Returns a string representation of the DE-9IM matrix of relationships between an object’s interior, boundary,exterior and those of another geometric object.
The named relationship predicates (contains(), etc.) are typically implemented as wrappers around relate().
Two different points have mainly F (false) values in their matrix; the intersection of their external sets (the 9th element)
is a 2 dimensional object (the rest of the plane). The intersection of the interior of one with the exterior of the other is
a 0 dimensional object (3rd and 7th elements of the matrix).
>>> Point(0, 0).relate(Point(1, 1))'FF0FFF0F2'
The matrix for a line and a point on the line has more “true” (not F) elements.
>>> Point(0, 0).relate(LineString([(0, 0), (1, 1)]))'F0FFFF102'
object.relate_pattern(other, pattern)
Returns True if the DE-9IM string code for the relationship between the geometries satisfies the pattern, otherwise
False.
The relate_pattern() compares the DE-9IM code string for two geometries against a specified pattern. If the string
matches the pattern then True is returned, otherwise False. The pattern specified can be an exact match (0, 1 or 2), aboolean match (T or F), or a wildcard (*). For example, the pattern for the within predicate is T*****FF*.

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Shapely Documentation, Release 1.8.1>> point = Point(0.5, 0.5)>> square = Polygon([(0, 0), (0, 1), (1, 1), (1, 0)])>> square.relate_pattern(point, 'T*****FF*')True>> point.within(square)
True
Note that the order or the geometries is significant, as demonstrated below. In this example the square contains thepoint, but the point does not contain the square.
>>> point.relate(square)'0FFFFF212'>>> square.relate(point)'0F2FF1FF2'
Further discussion of the DE-9IM matrix is beyond the scope of this manual. See? and https://pypi.org/project/de9im/.

1.2.4 Spatial Analysis Methods
As well as boolean attributes and methods, Shapely provides analysis methods that return new geometric objects.
Set-theoretic Methods
Almost every binary predicate method has a counterpart that returns a new geometric object. In addition, the settheoretic boundary of an object is available as a read-only attribute.
Note: These methods will always return a geometric object. An intersection of disjoint geometries for example will
return an empty GeometryCollection, not None or False. To test for a non-empty result, use the geometry’s is_empty
property.
object.boundary
Returns a lower dimensional object representing the object’s set-theoretic boundary.
The boundary of a polygon is a line, the boundary of a line is a collection of points. The boundary of a point is anempty (null) collection.
>> coords = [((0, 0), (1, 1)), ((-1, 0), (1, 0))]>>> lines = MultiLineString(coords)>>> lines.boundary<shapely.geometry.multipoint.MultiPoint object at 0x...>>>> pprint(list(lines.boundary))[<shapely.geometry.point.Point object at 0x...>,<shapely.geometry.point.Point object at 0x...>,<shapely.geometry.point.Point object at 0x...>,<shapely.geometry.point.Point object at 0x...>]>>> lines.boundary.boundary<shapely.geometry.collection.GeometryCollection object at 0x...>>>> lines.boundary.boundary.is_empty
True
See the figures in LineStrings and Collections of Lines for the illustration of lines and their boundaries.

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object.centroid
Returns a representation of the object’s geometric centroid (point).
>>> LineString([(0, 0), (1, 1)]).centroid<shapely.geometry.point.Point object at 0x...>>>> LineString([(0, 0), (1, 1)]).centroid.wkt'POINT (0.5000000000000000 0.5000000000000000)'
Note: The centroid of an object might be one of its points, but this is not guaranteed.
object.difference(other)
Returns a representation of the points making up this geometric object that do not make up the other object.
>>> a = Point(1, 1).buffer(1.5)>>> b = Point(2, 1).buffer(1.5)>>> a.difference(b)<shapely.geometry.polygon.Polygon object at 0x...>Note: The buffer() method is used to produce approximately circular polygons in the examples of this section; it
will be explained in detail later in this manual.

a.difference(b)

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Note: Shapely can not represent the difference between an object and a lower dimensional object (such as the difference
between a polygon and a line or point) as a single object, and in these cases the difference method returns a copy of the

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object named self.
object.intersection(other)
Returns a representation of the intersection of this object with the other geometric object.
>>> a = Point(1, 1).buffer(1.5)>>> b = Point(2, 1).buffer(1.5)>>> a.intersection(b)<shapely.geometry.polygon.Polygon object at 0x...>See the figure under symmetric_difference() below.
object.symmetric_difference(other)
Returns a representation of the points in this object not in the other geometric object, and the points in the othernot in this geometric object.
>>> a = Point(1, 1).buffer(1.5)>>> b = Point(2, 1).buffer(1.5)>>> a.symmetric_difference(b)<shapely.geometry.multipolygon.MultiPolygon object at ...>

a.intersection(b)

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object.union(other)
Returns a representation of the union of points from this object and the other geometric object.
The type of object returned depends on the relationship between the operands. The union of polygons (for example)
will be a polygon or a multi-polygon depending on whether they intersect or not.
>>> a = Point(1, 1).buffer(1.5)>>> b = Point(2, 1).buffer(1.5)(continues on next page)

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Shapely Documentation, Release 1.8.1(continued from previous page)>>> a.union(b)<shapely.geometry.polygon.Polygon object at 0x...>The semantics of these operations vary with type of geometric object. For example, compare the boundary of the unionof polygons to the union of their boundaries.
>>> a.union(b).boundary<shapely.geometry.polygon.LinearRing object at 0x...>>>> a.boundary.union(b.boundary)<shapely.geometry.multilinestring.MultiLineString object at 0x...>a.union(b)

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union() is an expensive way to find the cumulative union of many objects.
unary_union() for a more effective method.

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See shapely.ops.

Several of these set-theoretic methods can be invoked using overloaded operators:• intersection can be accessed with and, &• union can be accessed with or, |• difference can be accessed with minus, • symmetric_difference can be accessed with xor, ^1.2. The Shapely User Manual51

Shapely Documentation, Release 1.8.1>>> from shapely import wkt>>> p1 = wkt.loads('POLYGON((0 0, 1 0, 1 1, 0 1, 0 0))')>>> p2 = wkt.loads('POLYGON((0.5 0, 1.5 0, 1.5 1, 0.5 1, 0.5 0))')>>> (p1 & p2).wkt'POLYGON ((1 0, 0.5 0, 0.5 1, 1 1, 1 0))'>>> (p1 | p2).wkt'POLYGON ((0.5 0, 0 0, 0 1, 0.5 1, 1 1, 1.5 1, 1.5 0, 1 0, 0.5 0))'>>> (p1 - p2).wkt'POLYGON ((0.5 0, 0 0, 0 1, 0.5 1, 0.5 0))'>>> (p1 ^ p2).wkt'MULTIPOLYGON (((0.5 0, 0 0, 0 1, 0.5 1, 0.5 0)), ((1 0, 1 1, 1.5 1, 1.5 0, 1 0)))'Constructive Methods
Shapely geometric object have several methods that yield new objects not derived from set-theoretic analysis.
object.buffer(distance, resolution=16, cap_style=1, join_style=1, mitre_limit=5.0, single_sided=False)
Returns an approximate representation of all points within a given distance of the this geometric object.
The styles of caps are specified by integer values: 1 (round), 2 (flat), 3 (square). These values are also enumerated
by the object shapely.geometry.CAP_STYLE (see below).
The styles of joins between offset segments are specified by integer values: 1 (round), 2 (mitre), and 3 (bevel).
These values are also enumerated by the object shapely.geometry.JOIN_STYLE (see below).
shapely.geometry.CAP_STYLE
Attribute
round
flat
square

Value
1
2
3

Attribute
round
mitre
bevel

Value
1
2
3

shapely.geometry.JOIN_STYLE>>> from shapely.geometry import CAP_STYLE, JOIN_STYLE>>> CAP_STYLE.flat
2>>> JOIN_STYLE.bevel
3
A positive distance has an effect of dilation; a negative distance, erosion. The optional resolution argument determines
the number of segments used to approximate a quarter circle around a point.
>>> line = LineString([(0, 0), (1, 1), (0, 2), (2, 2), (3, 1), (1, 0)])>>> dilated = line.buffer(0.5)>>> eroded = dilated.buffer(-0.3)
Figure 9. Dilation of a line (left) and erosion of a polygon (right). New object is shown in blue.

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The default (resolution of 16) buffer of a point is a polygonal patch with 99.8% of the area of the circular disk it
approximates.
>>> p = Point(0, 0).buffer(10.0)>>> len(p.exterior.coords)
66>>> p.area
313.65484905459385
With a resolution of 1, the buffer is a square patch.
>>> q = Point(0, 0).buffer(10.0, 1)>>> len(q.exterior.coords)
5>>> q.area
200.0
You may want a buffer only on one side. You can achieve this effect with single_sided option.
The side used is determined by the sign of the buffer distance:• a positive distance indicates the left-hand side
• a negative distance indicates the right-hand side>>> line = LineString([(0, 0), (1, 1), (0, 2), (2, 2), (3, 1), (1, 0)])>>> left_hand_side = line.buffer(0.5, single_sided=True)>>> right_hand_side = line.buffer(-0.3, single_sided=True)
Figure 10. Single sided buffer of 0.5 left hand (left) and of 0.3 right hand (right).

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The single-sided buffer of point geometries is the same as the regular buffer. The End Cap Style for single-sided buffers
is always ignored, and forced to the equivalent of CAP_STYLE.flat.
Passed a distance of 0, buffer() can sometimes be used to “clean” self-touching or self-crossing polygons such as theclassic “bowtie”. Users have reported that very small distance values sometimes produce cleaner results than 0. Yourmileage may vary when cleaning surfaces.
>>> coords = [(0, 0), (0, 2), (1, 1), (2, 2), (2, 0), (1, 1), (0, 0)]>>> bowtie = Polygon(coords)>>> bowtie.is_valid
False>>> clean = bowtie.buffer(0)>>> clean.is_valid
True>>> clean<shapely.geometry.multipolygon.MultiPolygon object at ...>>>> len(clean.geoms)
2>>> list(clean.geoms[0].exterior.coords)[(0.0, 0.0), (0.0, 2.0), (1.0, 1.0), (0.0, 0.0)]>>> list(clean.geoms[1].exterior.coords)[(1.0, 1.0), (2.0, 2.0), (2.0, 0.0), (1.0, 1.0)]
Buffering splits the polygon in two at the point where they touch.
object.convex_hull
Returns a representation of the smallest convex Polygon containing all the points in the object unless the numberof points in the object is less than three. For two points, the convex hull collapses to a LineString; for 1, a Point.

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Shapely Documentation, Release 1.8.1>>> Point(0, 0).convex_hull<shapely.geometry.point.Point object at 0x...>>>> MultiPoint([(0, 0), (1, 1)]).convex_hull<shapely.geometry.linestring.LineString object at 0x...>>>> MultiPoint([(0, 0), (1, 1), (1, -1)]).convex_hull<shapely.geometry.polygon.Polygon object at 0x...>a) N = 2

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Figure 11. Convex hull (blue) of 2 points (left) and of 6 points (right).
object.envelope
Returns a representation of the point or smallest rectangular polygon (with sides parallel to the coordinate axes)
that contains the object.
>>> Point(0, 0).envelope<shapely.geometry.point.Point object at 0x...>>>> MultiPoint([(0, 0), (1, 1)]).envelope<shapely.geometry.polygon.Polygon object at 0x...>object.minimum_rotated_rectangle
Returns the general minimum bounding rectangle that contains the object. Unlike envelope this rectangle is notconstrained to be parallel to the coordinate axes. If the convex hull of the object is a degenerate (line or point)
this degenerate is returned.
New in Shapely 1.6.0>>> Point(0, 0).minimum_rotated_rectangle<shapely.geometry.point.Point object at 0x...>>>> MultiPoint([(0,0),(1,1),(2,0.5)]).minimum_rotated_rectangle<shapely.geometry.polygon.Polygon object at 0x...>1.2. The Shapely User Manual55

Shapely Documentation, Release 1.8.1a) MultiPoint

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b) LineString

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Figure 12. Minimum rotated rectangle for a multipoint feature (left) and a linestring feature (right).
object.parallel_offset(distance, side, resolution=16, join_style=1, mitre_limit=5.0)
Returns a LineString or MultiLineString geometry at a distance from the object on its right or its left side.
The distance parameter must be a positive float value.
The side parameter may be ‘left’ or ‘right’. Left and right are determined by following the direction of thegiven geometric points of the LineString. Right hand offsets are returned in the reverse direction of the original
LineString or LineRing, while left side offsets flow in the same direction.
The resolution of the offset around each vertex of the object is parameterized as in the buffer() method.
The join_style is for outside corners between line segments. Accepted integer values are 1 (round), 2 (mitre),and 3 (bevel). See also shapely.geometry.JOIN_STYLE.
Severely mitered corners can be controlled by the mitre_limit parameter (spelled in British English, en-gb). The
corners of a parallel line will be further from the original than most places with the mitre join style. The ratio of
this further distance to the specified distance is the miter ratio. Corners with a ratio which exceed the limit will
be beveled.
Note: This method may sometimes return a MultiLineString where a simple LineString was expected; forexample, an offset to a slightly curved LineString.

Note: This method is only available for LinearRing and LineString objects.
Figure 13. Three styles of parallel offset lines on the left side of a simple line string (its starting point shown as a circle)
and one offset on the right side, a multipart.

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The effect of the mitre_limit parameter is shown below.
Figure 14. Large and small mitre_limit values for left and right offsets.
object.simplify(tolerance, preserve_topology=True)
Returns a simplified representation of the geometric object.
All points in the simplified object will be within the tolerance distance of the original geometry. By default a slower
algorithm is used that preserves topology. If preserve topology is set to False the much quicker Douglas-Peucker
algorithm6 is used.
>>> p = Point(0.0, 0.0)>>> x = p.buffer(1.0)>>> x.area
3.1365484905459389>>> len(x.exterior.coords)
66>>> s = x.simplify(0.05, preserve_topology=False)>>> s.area
3.0614674589207187>>> len(s.exterior.coords)
17
Figure 15. Simplification of a nearly circular polygon using a tolerance of 0.2 (left) and 0.5 (right).
Note: Invalid geometric objects may result from simplification that does not preserve topology and simplification
may be sensitive to the order of coordinates: two geometries differing only in order of coordinates may be simplified
differently.

1.2.5 Affine Transformations
A collection of affine transform functions are in the shapely.affinity module, which return transformed geometries
by either directly supplying coefficients to an affine transformation matrix, or by using a specific, named transform(rotate, scale, etc.). The functions can be used with all geometry types (except GeometryCollection), and 3D types areeither preserved or supported by 3D affine transformations.
New in version 1.2.17.
shapely.affinity.affine_transform(geom, matrix)
Returns a transformed geometry using an affine transformation matrix.
The coefficient matrix is provided as a list or tuple with 6 or 12 items for 2D or 3D transformations, respectively.
For 2D affine transformations, the 6 parameter matrix is:[a, b, d, e, xoff, yoff]which represents the augmented matrix:⎡ ′⎤ ⎡𝑥𝑎 𝑏⎣ 𝑦 ′ ⎦ = ⎣𝑑 𝑒
1
0 0

⎤⎡ ⎤
𝑥off
𝑥
𝑦off ⎦ ⎣𝑦 ⎦1
1

6 David H. Douglas and Thomas K. Peucker, “Algorithms for the Reduction of the Number of Points Required to Represent a Digitized Line or
its Caricature,” Cartographica: The International Journal for Geographic Information and Geovisualization, vol. 10, Dec. 1973, pp. 112-122.

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or the equations for the transformed coordinates:𝑥′ = 𝑎𝑥 + 𝑏𝑦 + 𝑥off
𝑦 ′ = 𝑑𝑥 + 𝑒𝑦 + 𝑦off .
For 3D affine transformations, the 12 parameter matrix is:[a, b, c, d, e, f, g, h, i, xoff, yoff, zoff]
which represents the augmented matrix:⎡ ′⎤ ⎡𝑥𝑎⎢ 𝑦 ′ ⎥ ⎢𝑑⎢ ′⎥ = ⎢⎣ 𝑧 ⎦ ⎣𝑔
1
0

𝑏
𝑒
ℎ
0

𝑐
𝑓
𝑖
0

⎤⎡ ⎤
𝑥off
𝑥
⎢𝑦 ⎥
𝑦off ⎥⎥⎢ ⎥𝑧off ⎦ ⎣ 𝑧 ⎦1
1

or the equations for the transformed coordinates:𝑥′ = 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 + 𝑥off
𝑦 ′ = 𝑑𝑥 + 𝑒𝑦 + 𝑓 𝑧 + 𝑦off𝑧 ′ = 𝑔𝑥 + ℎ𝑦 + 𝑖𝑧 + 𝑧off .
shapely.affinity.rotate(geom, angle, origin='center', use_radians=False)
Returns a rotated geometry on a 2D plane.
The angle of rotation can be specified in either degrees (default) or radians by setting use_radians=True.
Positive angles are counter-clockwise and negative are clockwise rotations.
The point of origin can be a keyword 'center' for the bounding box center (default), 'centroid' for thegeometry’s centroid, a Point object or a coordinate tuple (x0, y0).
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The affine transformation matrix for 2D rotation with angle 𝜃 is:⎡⎤cos 𝜃 − sin 𝜃 𝑥off
⎣ sin 𝜃cos 𝜃 𝑦off ⎦001
where the offsets are calculated from the origin (𝑥0 , 𝑦0 ):𝑥off = 𝑥0 − 𝑥0 cos 𝜃 + 𝑦0 sin 𝜃𝑦off = 𝑦0 − 𝑥0 sin 𝜃 − 𝑦0 cos 𝜃>>> from shapely import affinity>>> line = LineString([(1, 3), (1, 1), (4, 1)])>>> rotated_a = affinity.rotate(line, 90)>>> rotated_b = affinity.rotate(line, 90, origin='centroid')90°, default origin (center)

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90°, origin='centroid'

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Figure 16. Rotation of a LineString (gray) by an angle of 90° counter-clockwise (blue) using different origins.
shapely.affinity.scale(geom, xfact=1.0, yfact=1.0, zfact=1.0, origin='center')
Returns a scaled geometry, scaled by factors along each dimension.
The point of origin can be a keyword 'center' for the 2D bounding box center (default), 'centroid' for thegeometry’s 2D centroid, a Point object or a coordinate tuple (x0, y0, z0).
Negative scale factors will mirror or reflect coordinates.
The general 3D affine transformation matrix for scaling is:
⎡
𝑥fact
0
0
⎢ 0
𝑦
0
fact
⎢
⎣ 0
0
𝑧fact
0
0
0
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where the offsets are calculated from the origin (𝑥0 , 𝑦0 , 𝑧0 ):𝑥off = 𝑥0 − 𝑥0 𝑥fact
𝑦off = 𝑦0 − 𝑦0 𝑦fact
𝑧off = 𝑧0 − 𝑧0 𝑧fact>>> triangle = Polygon([(1, 1), (2, 3), (3, 1)])>>> triangle_a = affinity.scale(triangle, xfact=1.5, yfact=-1)>>> triangle_a.exterior.coords[:][(0.5, 3.0), (2.0, 1.0), (3.5, 3.0), (0.5, 3.0)]>>> triangle_b = affinity.scale(triangle, xfact=2, origin=(1,1))>>> triangle_b.exterior.coords[:][(1.0, 1.0), (3.0, 3.0), (5.0, 1.0), (1.0, 1.0)]a) xfact=1.5, yfact=-1433(2.0, 2.0)

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4(1, 1)

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Figure 17. Scaling of a gray triangle to blue result: a) by a factor of 1.5 along x-direction, with reflection across
y-axis; b) by a factor of 2 along x-direction with custom origin at (1, 1).
shapely.affinity.skew(geom, xs=0.0, ys=0.0, origin='center', use_radians=False)
Returns a skewed geometry, sheared by angles along x and y dimensions.
The shear angle can be specified in either degrees (default) or radians by setting use_radians=True.
The point of origin can be a keyword 'center' for the bounding box center (default), 'centroid' for thegeometry’s centroid, a Point object or a coordinate tuple (x0, y0).
The general 2D affine transformation matrix for skewing is:⎡1
tan 𝑥𝑠⎣tan 𝑦𝑠
1
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0
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where the offsets are calculated from the origin (𝑥0 , 𝑦0 ):𝑥off = −𝑦0 tan 𝑥𝑠
𝑦off = −𝑥0 tan 𝑦𝑠a) xs=20, origin(1, 1)

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1001b) ys=30

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Figure 18. Skewing of a gray “R” to blue result: a) by a shear angle of 20° along the x-direction and an origin at(1, 1); b) by a shear angle of 30° along the y-direction, using default origin.
shapely.affinity.translate(geom, xoff=0.0, yoff=0.0, zoff=0.0)
Returns a translated geometry shifted by offsets along each dimension.
The general 3D affine transformation matrix for translation is:⎡⎤
1 0 0 𝑥off
⎢0 1 0 𝑦off ⎥⎢⎥⎣0 0 1 𝑧off ⎦0 0 0
1

1.2.6 Other Transformations
Shapely supports map projections and other arbitrary transformations of geometric objects.
shapely.ops.transform(func, geom)
Applies func to all coordinates of geom and returns a new geometry of the same type from the transformed
coordinates.
func maps x, y, and optionally z to output xp, yp, zp. The input parameters may be iterable types like lists orarrays or single values. The output shall be of the same type: scalars in, scalars out; lists in, lists out.

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transform tries to determine which kind of function was passed in by calling func first with n iterables of coordinates, where n is the dimensionality of the input geometry. If func raises a TypeError when called with iterables
as arguments, then it will instead call func on each individual coordinate in the geometry.
New in version 1.2.18.
For example, here is an identity function applicable to both types of input (scalar or array).
def id_func(x, y, z=None):return tuple(filter(None, [x, y, z]))g2 = transform(id_func, g1)
If using pyproj>=2.1.0, the preferred method to project geometries is:import pyproj
from shapely.geometry import Point
from shapely.ops import transform
wgs84_pt = Point(-72.2495, 43.886)
wgs84 = pyproj.CRS('EPSG:4326')
utm = pyproj.CRS('EPSG:32618')
project = pyproj.Transformer.from_crs(wgs84, utm, always_xy=True).transform
utm_point = transform(project, wgs84_pt)
It is important to note that in the example above, the always_xy kwarg is required as Shapely only supports coordinatesin X,Y order, and in PROJ 6 the WGS84 CRS uses the EPSG-defined Lat/Lon coordinate order instead of the expected
Lon/Lat.
If using pyproj < 2.1, then the canonical example is:from functools import partial
import pyproj
from shapely.ops import transform
wgs84 = pyproj.Proj(init='epsg:4326')
utm = pyproj.Proj(init='epsg:32618')
project = partial(
pyproj.transform,
wgs84,
utm)
utm_point = transform(project, wgs84_pt)
Lambda expressions such as the one in
g2 = transform(lambda x, y, z=None: (x+1.0, y+1.0), g1)
also satisfy the requirements for func.

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1.2.7 Other Operations
Merging Linear Features
Sequences of touching lines can be merged into MultiLineStrings or Polygons using functions in the shapely.ops
module.
shapely.ops.polygonize(lines)
Returns an iterator over polygons constructed from the input lines.
As with the MultiLineString constructor, the input elements may be any line-like object.
>>> from shapely.ops import polygonize>>> lines = [...
((0, 0), (1, 1)),...
((0, 0), (0, 1)),...
((0, 1), (1, 1)),...
((1, 1), (1, 0)),...
((1, 0), (0, 0))...
]>>> pprint(list(polygonize(lines)))[<shapely.geometry.polygon.Polygon object at 0x...>,<shapely.geometry.polygon.Polygon object at 0x...>]
shapely.ops.polygonize_full(lines)
Creates polygons from a source of lines, returning the polygons and leftover geometries.
The source may be a MultiLineString, a sequence of LineString objects, or a sequence of objects than can beadapted to LineStrings.
Returns a tuple of objects: (polygons, dangles, cut edges, invalid ring lines). Each are a geometry collection.
Dangles are edges which have one or both ends which are not incident on another edge endpoint. Cut edges areconnected at both ends but do not form part of polygon. Invalid ring lines form rings which are invalid (bowties,
etc).
New in version 1.2.18.
>>> lines = [...
((0, 0), (1, 1)),...
((0, 0), (0, 1)),...
((0, 1), (1, 1)),...
((1, 1), (1, 0)),...
((1, 0), (0, 0)),...
((5, 5), (6, 6)),...
((1, 1), (100, 100)),...
]>>> result, dangles, cuts, invalids = polygonize_full(lines)>>> len(result.geoms)
2>>> list(result.geoms)[<shapely.geometry.polygon.Polygon object at ...>, <shapely.geometry.polygon.
˓→Polygon object at ...>]>>> list(cuts.geoms)[<shapely.geometry.linestring.LineString object at ...>, <shapely.geometry.
˓→linestring.LineString object at ...>]

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shapely.ops.linemerge(lines)
Returns a LineString or MultiLineString representing the merger of all contiguous elements of lines.
As with shapely.ops.polygonize(), the input elements may be any line-like object.
>>> from shapely.ops import linemerge>>> linemerge(lines)<shapely.geometry.multilinestring.MultiLineString object at 0x...>>>> pprint(list(linemerge(lines)))[<shapely.geometry.linestring.LineString object at 0x...>,<shapely.geometry.linestring.LineString object at 0x...>,<shapely.geometry.linestring.LineString object at 0x...>]

Efficient Rectangle Clipping
The clip_by_rect() function in shapely.ops returns the portion of a geometry within a rectangle.
shapely.ops.clip_by_rect(geom, xmin, ymin, xmax, ymax)
The geometry is clipped in a fast but possibly dirty way. The output is not guaranteed to be valid. No exceptions
will be raised for topological errors.
New in version 1.7.
Requires GEOS 3.5.0 or higher>>> from shapely.geometry import Polygon>>> from shapely.ops import clip_by_rect>>> polygon = Polygon(
shell=[(0, 0), (0, 30), (30, 30), (30, 0), (0, 0)],holes=[[(10, 10), (20, 10), (20, 20), (10, 20), (10, 10)]],)>>> clipped_polygon = clip_by_rect(polygon, 5, 5, 15, 15)>>> print(clipped_polygon.wkt)
POLYGON ((5 5, 5 15, 10 15, 10 10, 15 10, 15 5, 5 5))

Efficient Unions
The unary_union() function in shapely.ops is more efficient than accumulating with union().
shapely.ops.unary_union(geoms)
Returns a representation of the union of the given geometric objects.
Areas of overlapping Polygons will get merged. LineStrings will get fully dissolved and noded. Duplicate Points
will get merged.
>>> from shapely.ops import unary_union>>> polygons = [Point(i, 0).buffer(0.7) for i in range(5)]>>> unary_union(polygons)<shapely.geometry.polygon.Polygon object at 0x...>Because the union merges the areas of overlapping Polygons it can be used in an attempt to fix invalid MultiPolygons. As with the zero distance buffer() trick, your mileage may vary when using this.

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>>> m = MultiPolygon(polygons)>>> m.area
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False>>> unary_union(m).area
6.6103013551167971>>> unary_union(m).is_valid
Trueshapely.ops.cascaded_union(geoms)
Returns a representation of the union of the given geometric objects.
Note: In 1.8.0 shapely.ops.cascaded_union() is deprecated, as it was superseded by shapely.ops.
unary_union().

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Delaunay triangulation
The triangulate() function in shapely.ops calculates a Delaunay triangulation from a collection of points.

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shapely.ops.triangulate(geom, tolerance=0.0, edges=False)
Returns a Delaunay triangulation of the vertices of the input geometry.
The source may be any geometry type. All vertices of the geometry will be used as the points of the triangulation.
The tolerance keyword argument sets the snapping tolerance used to improve the robustness of the triangulation
computation. A tolerance of 0.0 specifies that no snapping will take place.
If the edges keyword argument is False a list of Polygon triangles will be returned. Otherwise a list of LineStringedges is returned.
New in version 1.4.0>>> from shapely.ops import triangulate>>> points = MultiPoint([(0, 0), (1, 1), (0, 2), (2, 2), (3, 1), (1, 0)])>>> triangles = triangulate(points)>>> pprint([triangle.wkt for triangle in triangles])['POLYGON ((0 2, 0 0, 1 1, 0 2))','POLYGON ((0 2, 1 1, 2 2, 0 2))','POLYGON ((2 2, 1 1, 3 1, 2 2))','POLYGON ((3 1, 1 1, 1 0, 3 1))','POLYGON ((1 0, 1 1, 0 0, 1 0))']68

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Voronoi Diagram
The voronoi_diagram() function in shapely.ops constructs a Voronoi diagram from a collection points, or the vertices of any geometry.

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shapely.ops.voronoi_diagram(geom, envelope=None, tolerance=0.0, edges=False)
Constructs a Voronoi diagram from the vertices of the input geometry.
The source may be any geometry type. All vertices of the geometry will be used as the input points to the diagram.
The envelope keyword argument provides an envelope to use to clip the resulting diagram. If None, it will becalculated automatically. The diagram will be clipped to the larger of the provided envelope or an envelope
surrounding the sites.
The tolerance keyword argument sets the snapping tolerance used to improve the robustness of the computation.
A tolerance of 0.0 specifies that no snapping will take place. The tolerance argument can be finicky and is known
to cause the algorithm to fail in several cases. If you’re using tolerance and getting a failure, try removing it. The
test cases in tests/test_voronoi_diagram.py show more details.
If the edges keyword argument is False a list of Polygon`s will be returned. Otherwise a list of `LineString edges
is returned.
>>> from shapely.ops import voronoi_diagram>>> points = MultiPoint([(0, 0), (1, 1), (0, 2), (2, 2), (3, 1), (1, 0)])>>> regions = voronoi_diagram(points)>>> pprint([region.wkt for region in regions])['POLYGON ((2 1, 2 0.5, 0.5 0.5, 0 1, 1 2, 2 1))','POLYGON ((6 5, 6 -3, 3.75 -3, 2 0.5, 2 1, 6 5))','POLYGON ((0.5 -3, -3 -3, -3 1, 0 1, 0.5 0.5, 0.5 -3))','POLYGON ((3.75 -3, 0.5 -3, 0.5 0.5, 2 0.5, 3.75 -3))',(continues on next page)1.2. The Shapely User Manual69

Shapely Documentation, Release 1.8.1(continued from previous page)'POLYGON ((-3 1, -3 5, 1 5, 1 2, 0 1, -3 1))','POLYGON ((1 5, 6 5, 2 1, 1 2, 1 5))']

Nearest points
The nearest_points() function in shapely.ops calculates the nearest points in a pair of geometries.
shapely.ops.nearest_points(geom1, geom2)
Returns a tuple of the nearest points in the input geometries. The points are returned in the same order as the
input geometries.
New in version 1.4.0.
>>> from shapely.ops import nearest_points>>> triangle = Polygon([(0, 0), (1, 0), (0.5, 1), (0, 0)])>>> square = Polygon([(0, 2), (1, 2), (1, 3), (0, 3), (0, 2)])>>> [o.wkt for o in nearest_points(triangle, square)]['POINT (0.5 1)', 'POINT (0.5 2)']
Note that the nearest points may not be existing vertices in the geometries.
Snapping
The snap() function in shapely.ops snaps the vertices in one geometry to the vertices in a second geometry with agiven tolerance.
shapely.ops.snap(geom1, geom2, tolerance)
Snaps vertices in geom1 to vertices in the geom2. A copy of the snapped geometry is returned. The input
geometries are not modified.
The tolerance argument specifies the minimum distance between vertices for them to be snapped.
New in version 1.5.0>>> from shapely.ops import snap>>> square = Polygon([(1,1), (2, 1), (2, 2), (1, 2), (1, 1)])>>> line = LineString([(0,0), (0.8, 0.8), (1.8, 0.95), (2.6, 0.5)])>>> result = snap(line, square, 0.5)>>> result.wkt'LINESTRING (0 0, 1 1, 2 1, 2.6 0.5)'

Shared paths
The shared_paths() function in shapely.ops finds the shared paths between two linear geometries.
shapely.ops.shared_paths(geom1, geom2)
Finds the shared paths between geom1 and geom2, where both geometries are LineStrings.
A GeometryCollection is returned with two elements. The first element is a MultiLineString containing sharedpaths with the same direction for both inputs. The second element is a MultiLineString containing shared paths
with the opposite direction for the two inputs.
New in version 1.6.0

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Shapely Documentation, Release 1.8.1>>> from shapely.ops import shared_paths>>> g1 = LineString([(0, 0), (10, 0), (10, 5), (20, 5)])>>> g2 = LineString([(5, 0), (30, 0), (30, 5), (0, 5)])>>> forward, backward = shared_paths(g1, g2)>>> forward.wkt'MULTILINESTRING ((5 0, 10 0))'>>> backward.wkt'MULTILINESTRING ((10 5, 20 5))'

Splitting
The split() function in shapely.ops splits a geometry by another geometry.
shapely.ops.split(geom, splitter)
Splits a geometry by another geometry and returns a collection of geometries. This function is the theoretical
opposite of the union of the split geometry parts. If the splitter does not split the geometry, a collection with asingle geometry equal to the input geometry is returned.
The function supports:• Splitting a (Multi)LineString by a (Multi)Point or (Multi)LineString or (Multi)Polygon boundary• Splitting a (Multi)Polygon by a LineString
It may be convenient to snap the splitter with low tolerance to the geometry. For example in the case of splitting
a line by a point, the point must be exactly on the line, for the line to be correctly split. When splitting a line
by a polygon, the boundary of the polygon is used for the operation. When splitting a line by another line, aValueError is raised if the two overlap at some segment.
New in version 1.6.0>>> pt = Point((1, 1))>>> line = LineString([(0,0), (2,2)])>>> result = split(line, pt)>>> result.wkt'GEOMETRYCOLLECTION (LINESTRING (0 0, 1 1), LINESTRING (1 1, 2 2))'

Substring
The substring() function in shapely.ops returns a line segment between specified distances along a LineString.
shapely.ops.substring(geom, start_dist, end_dist [, normalized=False ])Return the LineString between start_dist and end_dist or a Point if they are at the same location
Negative distance values are taken as measured in the reverse direction from the end of the geometry. Out-ofrange index values are handled by clamping them to the valid range of values.
If the start distance equals the end distance, a point is being returned.
If the start distance is actually past the end distance, then the reversed substring is returned such that the start
distance is at the first coordinate.
If the normalized arg is True, the distance will be interpreted as a fraction of the geometry’s length
New in version 1.7.0
Here are some examples that return LineString geometries.

1.2. The Shapely User Manual71Shapely Documentation, Release 1.8.1>>> from shapely.geometry import LineString>>> from shapely.ops import substring>>> ls = LineString((i, 0) for i in range(6))>>> ls.wkt'LINESTRING (0 0, 1 0, 2 0, 3 0, 4 0, 5 0)'>>> substring(ls, start_dist=1, end_dist=3).wkt'LINESTRING (1 0, 2 0, 3 0)'>>> substring(ls, start_dist=3, end_dist=1).wkt'LINESTRING (3 0, 2 0, 1 0)'>>> substring(ls, start_dist=1, end_dist=-3).wkt'LINESTRING (1 0, 2 0)'>>> substring(ls, start_dist=0.2, end_dist=-0.6, normalized=True).wkt'LINESTRING (1 0, 2 0)'
And here is an example that returns a Point.
>>> substring(ls, start_dist=2.5, end_dist=-2.5)'POINT (2.5 0)'

Prepared Geometry Operations
Shapely geometries can be processed into a state that supports more efficient batches of operations.
prepared.prep(ob)Creates and returns a prepared geometric object.
To test one polygon containment against a large batch of points, one should first use the prepared.prep() function.
>>> from shapely.geometry import Point>>> from shapely.prepared import prep>>> points = [...] # large list of points>>> polygon = Point(0.0, 0.0).buffer(1.0)>>> prepared_polygon = prep(polygon)>>> prepared_polygon<shapely.prepared.PreparedGeometry object at 0x...>>>> hits = filter(prepared_polygon.contains, points)
Prepared geometries instances have the following methods: contains, contains_properly, covers, andintersects. All have exactly the same arguments and usage as their counterparts in non-prepared geometric objects.
Diagnostics
validation.explain_validity(ob):Returns a string explaining the validity or invalidity of the object.
New in version 1.2.1.
The messages may or may not have a representation of a problem point that can be parsed out.
>>> coords = [(0, 0), (0, 2), (1, 1), (2, 2), (2, 0), (1, 1), (0, 0)]>>> p = Polygon(coords)>>> from shapely.validation import explain_validity(continues on next page)

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Shapely Documentation, Release 1.8.1(continued from previous page)>>> explain_validity(p)'Ring Self-intersection[1 1]'
validation.make_valid(ob)Returns a valid representation of the geometry, if it is invalid. If it is valid, the input geometry will be returned.
In many cases, in order to create a valid geometry, the input geometry must be split into multiple parts or multiple
geometries. If the geometry must be split into multiple parts of the same geometry type, then a multi-partgeometry (e.g. a MultiPolygon) will be returned. if the geometry must be split into multiple parts of different
types, then a GeometryCollection will be returned.
For example, this operation on a geometry with a bow-tie structure:>>> from shapely.validation import make_valid>>> coords = [(0, 0), (0, 2), (1, 1), (2, 2), (2, 0), (1, 1), (0, 0)]>>> p = Polygon(coords)>>> str(make_valid(p))'MULTIPOLYGON (((0 0, 0 2, 1 1, 0 0)), ((1 1, 2 2, 2 0, 1 1)))'
Yields a MultiPolygon with two parts:

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Fig. 1: While this operation:>>> from shapely.validation import make_valid>>> coords = [(0, 2), (0, 1), (2, 0), (0, 0), (0, 2)]>>> p = Polygon(coords)>>> str(make_valid(p))(continues on next page)1.2. The Shapely User Manual73Shapely Documentation, Release 1.8.1(continued from previous page)

Yields a GeometryCollection with a Polygon and a LineString:

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The Shapely version, GEOS library version, and GEOS C API version are accessible via shapely.__version__,shapely.geos.geos_version_string, and shapely.geos.geos_capi_version.
>>> import shapely>>> shapely.__version__'1.3.0'>>> import shapely.geos>>> shapely.geos.geos_version(3, 3, 0)>>> shapely.geos.geos_version_string'3.3.0-CAPI-1.7.0'

Polylabelshapely.ops.polylabel(polygon, tolerance)
Finds the approximate location of the pole of inaccessibility for a given polygon. Based on Vladimir Agafonkin’s
polylabel.
New in version 1.6.0
Note: Prior to 1.7 polylabel must be imported from shapely.algorithms.polylabel instead of shapely.ops.

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Shapely Documentation, Release 1.8.1>>> from shapely.ops import polylabel>>> polygon = LineString([(0, 0), (50, 200), (100, 100), (20, 50),... (-100, -20), (-150, -200)]).buffer(100)>>> label = polylabel(polygon, tolerance=10)>>> label.wkt'POINT (59.35615556364569 121.8391962974644)'

1.2.8 STR-packed R-tree
Shapely provides an interface to the query-only GEOS R-tree packed using the Sort-Tile-Recursive algorithm. Pass alist of geometry objects to the STRtree constructor to create a spatial index that you can query with another geometricobject. Query-only means that once created, the STRtree is immutable. You cannot add or remove geometries.
class strtree.STRtree(geometries)
The STRtree constructor takes a sequence of geometric objects.
References to these geometric objects are kept and stored in the R-tree.
New in version 1.4.0.
strtree.query(geom)
Returns a list of all geometries in the strtree whose extents intersect the extent of geom. This means thata subsequent search through the returned subset using the desired binary predicate (eg. intersects, crosses,
contains, overlaps) may be necessary to further filter the results according to their specific spatial relationships.
>>> from shapely.strtree import STRtree>>> points = [Point(i, i) for i in range(10)]>>> tree = STRtree(points)>>> query_geom = Point(2,2).buffer(0.99)>>> [o.wkt for o in tree.query(query_geom)]['POINT (2 2)']>>> query_geom = Point(2, 2).buffer(1.0)>>> [o.wkt for o in tree.query(query_geom)]['POINT (1 1)', 'POINT (2 2)', 'POINT (3 3)']>>> [o.wkt for o in tree.query(query_geom) if o.intersects(query_geom)]['POINT (2 2)']

Note: To get the original indexes of the query results, create an auxiliary dictionary. But use the geometry
ids as keys since the shapely geometries themselves are not hashable.
>>> index_by_id = dict((id(pt), i) for i, pt in enumerate(points))>>> [(index_by_id[id(pt)], pt.wkt) for pt in tree.query(Point(2,2).buffer(1.0))][(1, 'POINT (1 1)'), (2, 'POINT (2 2)'), (3, 'POINT (3 3)')]strtree.nearest(geom)
Returns the nearest geometry in strtree to geom.
>>> tree = STRtree([Point(i, i) for i in range(10)])>>> tree.nearest(Point(2.2, 2.2)).wkt'Point (2 2)'

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1.2.9 Interoperation
Shapely provides 4 avenues for interoperation with other software.
Well-Known Formats
A Well Known Text (WKT) or Well Known Binary (WKB) representation? of any geometric object can be had via its
wkt or wkb attribute. These representations allow interchange with many GIS programs. PostGIS, for example, tradesin hex-encoded WKB.
>>> Point(0, 0).wkt'POINT (0.0000000000000000 0.0000000000000000)'>>> Point(0, 0).wkb.encode('hex')'010100000000000000000000000000000000000000'
The shapely.wkt and shapely.wkb modules provide dumps() and loads() functions that work almost exactly as their
pickle and simplejson module counterparts. To serialize a geometric object to a binary or text string, use dumps(). To
deserialize a string and get a new geometric object of the appropriate type, use loads().
The default settings for the wkt attribute and shapely.wkt.dumps() function are different. By default, the attribute’s value
is trimmed of excess decimals, while this is not the case for dumps(), though it can be replicated by setting trim=True.
shapely.wkb.dumps(ob)Returns a WKB representation of ob.
shapely.wkb.loads(wkb)
Returns a geometric object from a WKB representation wkb.
>>> from shapely import wkb>>> pt = Point(0, 0)>>> wkb.dumps(pt)b'\x01\x01\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00'>>> pt.wkb
b'\x01\x01\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00'>>> pt.wkb_hex'010100000000000000000000000000000000000000'>>> wkb.loads(pt.wkb).wkt'POINT (0 0)'
All of Shapely’s geometry types are supported by these functions.
shapely.wkt.dumps(ob)Returns a WKT representation of ob. Several keyword arguments are available to alter the WKT which is returned; see the docstrings for more details.
shapely.wkt.loads(wkt)Returns a geometric object from a WKT representation wkt.
>>> from shapely import wkt>>> pt = Point(0, 0)>>> thewkt = wkt.dumps(pt)>>> thewkt'POINT (0.0000000000000000 0.0000000000000000)'>>> pt.wkt'POINT (0 0)'(continues on next page)

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Shapely Documentation, Release 1.8.1(continued from previous page)>>> wkt.dumps(pt, trim=True)'POINT (0 0)'

Numpy and Python Arrays
All geometric objects with coordinate sequences (Point, LinearRing, LineString) provide the Numpy array interface
and can thereby be converted or adapted to Numpy arrays.
>>> from numpy import asarray>>> asarray(Point(0, 0))array([ 0., 0.])>>> asarray(LineString([(0, 0), (1, 1)]))array([[ 0., 0.],[ 1., 1.]])

Note: The Numpy array interface is provided without a dependency on Numpy itself.
The coordinates of the same types of geometric objects can be had as standard Python arrays of x and y values via thexy attribute.
>>> Point(0, 0).xy(array('d', [0.0]), array('d', [0.0]))>>> LineString([(0, 0), (1, 1)]).xy(array('d', [0.0, 1.0]), array('d', [0.0, 1.0]))
The shapely.geometry.asShape() family of functions can be used to wrap Numpy coordinate arrays so that they
can then be analyzed using Shapely while maintaining their original storage. A 1 x 2 array can be adapted to a point>>> from shapely.geometry import asPoint>>> pa = asPoint(array([0.0, 0.0]))>>> pa.wkt'POINT (0.0000000000000000 0.0000000000000000)'
and a N x 2 array can be adapted to a line string>>> from shapely.geometry import asLineString>>> la = asLineString(array([[1.0, 2.0], [3.0, 4.0]]))>>> la.wkt'LINESTRING (1.0000000000000000 2.0000000000000000, 3.0000000000000000 4.
˓→0000000000000000)'
Polygon and MultiPoint can also be created from N x 2 arrays:>>> from shapely.geometry import asMultiPoint>>> ma = asMultiPoint(np.array([[1.1, 2.2], [3.3, 4.4], [5.5, 6.6]]))>>> ma.wkt'MULTIPOINT (1.1 2.2, 3.3 4.4, 5.5 6.6)'>>> from shapely.geometry import asPolygon>>> pa = asPolygon(np.array([[1.1, 2.2], [3.3, 4.4], [5.5, 6.6]]))(continues on next page)1.2. The Shapely User Manual77

Shapely Documentation, Release 1.8.1(continued from previous page)>>> pa.wkt'POLYGON ((1.1 2.2, 3.3 4.4, 5.5 6.6, 1.1 2.2))'

Python Geo Interface
Any object that provides the GeoJSON-like Python geo interface can be adapted and used as a Shapely geometry using
the shapely.geometry.asShape() or shapely.geometry.shape() functions.
shapely.geometry.asShape(context)
Adapts the context to a geometry interface. The coordinates remain stored in the context.
shapely.geometry.shape(context)
Returns a new, independent geometry with coordinates copied from the context.
For example, a dictionary:>>> from shapely.geometry import shape>>> data = {"type": "Point", "coordinates": (0.0, 0.0)}>>> geom = shape(data)>>> geom.geom_type'Point'>>> list(geom.coords)[(0.0, 0.0)]
Or a simple placemark-type object:>>> class GeoThing:...
def __init__(self, d):...
self.__geo_interface__ = d>>> thing = GeoThing({"type": "Point", "coordinates": (0.0, 0.0)})>>> geom = shape(thing)>>> geom.geom_type'Point'>>> list(geom.coords)[(0.0, 0.0)]
The GeoJSON-like mapping of a geometric object can be obtained using shapely.geometry.mapping().
shapely.geometry.mapping(ob)Returns a new, independent geometry with coordinates copied from the context.
New in version 1.2.3.
For example, using the same GeoThing class:>>> from shapely.geometry import mapping>>> thing = GeoThing({"type": "Point", "coordinates": (0.0, 0.0)})>>> m = mapping(thing)>>> m['type']'Point'>>> m['coordinates'](0.0, 0.0)}78

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1.2.10 Performance
Shapely uses the GEOS library for all operations. GEOS is written in C++ and used in many applications and you
can expect that all operations are highly optimized. The creation of new geometries with many coordinates, however,involves some overhead that might slow down your code.
New in version 1.2.10.
The shapely.speedups module contains performance enhancements written in C. They are automatically installed
when Python has access to a compiler and GEOS development headers during installation.
You can check if the speedups are installed with the available attribute. To enable the speedups call enable(). You
can revert to the slow implementation with disable().
>>> from shapely import speedups>>> speedups.available
True>>> speedups.enable()
New in version 1.6.0.
Speedups are now enabled by default if they are available. You can check if speedups are enabled with the enabledattribute.
>>> from shapely import speedups>>> speedups.enabledTrue

1.2.11 Conclusion
We hope that you will enjoy and profit from using Shapely. This manual will be updated and improved regularly. Its
source is available at https://github.com/shapely/shapely/tree/maint-1.8/docs/.

1.2.12 References

1.3 Migrating to Shapely 1.8 / 2.0
Shapely 1.8.0 is a transitional version introducing several warnings in preparation of the upcoming changes in 2.0.0.
Shapely 2.0.0 will be a major release with a refactor of the internals with considerable performance improvements(based on the developments in the PyGEOS package), along with several breaking changes.
This guide gives an overview of the most important changes with details on what will change in 2.0.0, how we warn
for this in 1.8.0, and how you can update your code to be future-proof.
For more background, see RFC 1: Roadmap for Shapely 2.0.
Table of Contents• Geometry objects will become immutable– Setting custom attributes• Multi-part geometries will no longer be “sequences” (length, iterable, indexable)

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1.3.1 Geometry objects will become immutable
Geometry objects will become immutable in version 2.0.0.
In Shapely 1.x, some of the geometry classes are mutable, meaning that you can change their coordinates in-place.
Illustrative code:>>> from shapely.geometry import LineString>>> line = LineString([(0,0), (2, 2)])>>> print(line)
LINESTRING (0 0, 2 2)>>> line.coords = [(0, 0), (10, 0), (10, 10)]>>> print(line)
LINESTRING (0 0, 10 0, 10 10)
In Shapely 1.8, this will start raising a warning:>>> line.coords = [(0, 0), (10, 0), (10, 10)]
ShapelyDeprecationWarning: Setting the 'coords' to mutate a Geometry
in place is deprecated, and will not be possible any more in Shapely 2.0
and starting with version 2.0.0, all geometry objects will become immutable. As a consequence, they will also becomehashable and therefore usable as, for example, dictionary keys.
How do I update my code? There is no direct alternative for mutating the coordinates of an existing geometry, except
for creating a new geometry object with the new coordinates.
Setting custom attributes
Another consequence of the geometry objects becoming immutable is that assigning custom attributes, which currently
works, will no longer be possible.
Currently you can do:>>> line.name = "my_geometry">>> line.name'my_geometry'
In Shapely 1.8, this will start raising a warning, and will raise an AttributeError in Shapely 2.0.
How do I update my code? There is no direct alternative for adding custom attributes to geometry objects. You can
use other Python data structures such as (GeoJSON-like) dictionaries or GeoPandas’ GeoDataFrames to store attributes
alongside geometry features.

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1.3.2 Multi-part geometries will no longer be “sequences” (length, iterable, indexable)
In Shapely 1.x, multi-part geometries (MultiPoint, MultiLineString, MultiPolygon and GeometryCollection) implement a part of the “sequence” python interface (making them list-like). This means you can iterate through the object
to get the parts, index into the object to get a specific part, and ask for the number of parts with the len() method.
Some examples of this with Shapely 1.x:>>> from shapely.geometry import Point, MultiPoint>>> mp = MultiPoint([(1, 1), (2, 2), (3, 3)])>>> print(mp)
MULTIPOINT (1 1, 2 2, 3 3)>>> for part in mp:...
print(part)
POINT (1 1)
POINT (2 2)POINT (3 3)>>> print(mp[1])POINT (2 2)>>> len(mp)3>>> list(mp)[<shapely.geometry.point.Point at 0x7f2e0912bf10>,<shapely.geometry.point.Point at 0x7f2e09fed820>,<shapely.geometry.point.Point at 0x7f2e09fed4c0>]
Starting with Shapely 1.8, all the examples above will start raising a deprecation warning. For example:>>> for part in mp:...
print(part)
ShapelyDeprecationWarning: Iteration over multi-part geometries is deprecatedand will be removed in Shapely 2.0. Use the `geoms` property to access theconstituent parts of a multi-part geometry.
POINT (1 1)
POINT (2 2)POINT (3 3)
In Shapely 2.0, all those examples will raise an error.
How do I update my code? To access the geometry parts of a multi-part geometry, you can use the .geoms attribute,as the warning indicates.
The examples above can be updated to:>>> for part in mp.geoms:...
print(part)
POINT (1 1)
POINT (2 2)POINT (3 3)>>> print(mp.geoms[1])POINT (2 2)>>> len(mp.geoms)
3>>> list(mp.geoms)(continues on next page)

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The single-part geometries (Point, LineString, Polygon) already didn’t support those features, and for those classes
there is no change in behaviour for this aspect.

1.3.3 Interopability with NumPy and the array interface
Conversion of the coordinates to (NumPy) arrays
Shapely provides an array interface to have easy access to the coordinates as, for example, NumPy arrays (manual
section).
A small example:>>> line = LineString([(0, 0), (1, 1), (2, 2)])>>> import numpy as np>>> np.asarray(line)
array([[0., 0.],[1., 1.],[2., 2.]])
In addition, there are also the explicit array_interface() method and ctypes attribute to get access to the coordinates as array data:>>> line.ctypes<shapely.geometry.linestring.c_double_Array_6 at 0x7f75261eb740>>>> line.array_interface(){'version': 3,'typestr': '<f8','data': <shapely.geometry.linestring.c_double_Array_6 at 0x7f752664ae40>,'shape': (3, 2)}This functionality is available for Point, LineString, LinearRing and MultiPoint.
For more robust interoperability with NumPy, this array interface will be removed from those geometry classes, andlimited to the coords.
Starting with Shapely 1.8, converting a geometry object to a NumPy array directly will start raising a warning:>>> np.asarray(line)
ShapelyDeprecationWarning: The array interface is deprecated and will no longer
work in Shapely 2.0. Convert the '.coords' to a NumPy array instead.
array([[0., 0.],[1., 1.],[2., 2.]])How do I update my code? To convert a geometry to a NumPy array, you can convert the .coords attribute instead:>>> line.coords<shapely.coords.CoordinateSequence at 0x7f2e09e88d60>>>> np.array(line.coords)(continues on next page)

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Shapely Documentation, Release 1.8.1(continued from previous page)array([[0., 0.],[1., 1.],[2., 2.]])
The array_interface() method and ctypes attribute will be removed in Shapely 2.0, but since Shapely will startrequiring NumPy as a dependency, you can use NumPy or its array interface directly. Check the NumPy docs on thectypes attribute or the array interface for more details.
Creating NumPy arrays of geometry objects
Shapely geometry objects can be stored in NumPy arrays using the object dtype. In general, one could create suchan array from a list of geometries as follows:>>> from shapely.geometry import Point>>> arr = np.array([Point(0, 0), Point(1, 1), Point(2, 2)])>>> arrarray([<shapely.geometry.point.Point object at 0x7fb798407cd0>,<shapely.geometry.point.Point object at 0x7fb7982831c0>,<shapely.geometry.point.Point object at 0x7fb798283b80>],
dtype=object)
The above works for point geometries, but because in Shapely 1.x, some geometry types are sequence-like (see above),
NumPy can try to “unpack” them when creating an array. Therefore, for more robust creation of a NumPy array froma list of geometries, it’s generally recommended to this in a two-step way (first creating an empty array and then fillingit):geoms = [Point(0, 0), Point(1, 1), Point(2, 2)]
arr = np.empty(len(geoms), dtype="object")arr[:] = geoms
This code snippet results in the same array as the example above, and works for all geometry types and Shapely/NumPy
versions.
However, starting with Shapely 1.8, the above code will show deprecation warnings that cannot be avoided (dependingon the geometry type, NumPy tries to access the array interface of the objects or check if an object is iterable or has alength, and those operations are all deprecated now. The end result is still correct, but the warnings appear nonetheless).
Specifically in this case, it is fine to ignore those warnings (and the only way to make them go away):
import warnings
from shapely.errors import ShapelyDeprecationWarning
geoms = [Point(0, 0), Point(1, 1), Point(2, 2)]
arr = np.empty(len(geoms), dtype="object")with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=ShapelyDeprecationWarning)arr[:] = geoms
In Shapely 2.0, the geometry objects will no longer be sequence like and those deprecation warnings will be removed(and thus the filterwarnings will no longer be necessary), and creation of NumPy arrays will generally be morerobust.
If you maintain code that depends on Shapely, and you want to have it work with multiple versions of Shapely, theabove code snippet provides a context manager that can be copied into your project:
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import contextlib
import shapely
import warnings
from distutils.version import LooseVersion
SHAPELY_GE_20 = str(shapely.__version__) >= LooseVersion("2.0")
try:from shapely.errors import ShapelyDeprecationWarning as shapely_warning
except ImportError:
shapely_warning = Noneif shapely_warning is not None and not SHAPELY_GE_20:@contextlib.contextmanager
def ignore_shapely2_warnings():
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=shapely_warning)
yield
else:
@contextlib.contextmanager
def ignore_shapely2_warnings():
yield
This can then be used when creating NumPy arrays (be careful to only use it for this specific purpose, and not generally
suppress those warnings):
geoms = [...]
arr = np.empty(len(geoms), dtype="object")with ignore_shapely2_warnings():arr[:] = geoms

1.3.4 Consistent creation of empty geometries
Shapely 1.x is inconsistent in creating empty geometries between various creation methods. A small example for an
empty Polygon geometry:# Using an empty constructor results in a GeometryCollection>>> from shapely.geometry import Polygon>>> g1 = Polygon()>>> type(g1)<class 'shapely.geometry.polygon.Polygon'>>>> g1.wkt
GEOMETRYCOLLECTION EMPTY# Converting from WKT gives a correct empty polygon>>> from shapely import wkt>>> g2 = wkt.loads("POLYGON EMPTY")>>> type(g2)<class 'shapely.geometry.polygon.Polygon'>>>> g2.wkt
POLYGON EMPTY

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Shapely 1.8 does not yet change this inconsistent behaviour, but starting with Shapely 2.0, the different methods will
always consistently give an empty geometry object of the correct type, instead of using an empty GeometryCollection
as “generic” empty geometry object.
How do I update my code? Those cases that will change don’t raise a warning, but you will need to update your code
if you rely on the fact that empty geometry objects are of the GeometryCollection type. Use the .is_empty attributefor robustly checking if a geometry object is an empty geometry.
In addition, the WKB serialization methods will start supporting empty Points (using "POINT (NaN NaN)" to represent an empty point).

1.3.5 Other deprecated functionality
There are some other various functions and methods deprecated in Shapely 1.8 as well:• The adapters to create geometry-like proxy objects with coordinates stored outside Shapely geometries are deprecated and will be removed in Shapely 2.0 (e.g. created using asShape()). They have little to no benefit comparedto the normal geometry classes, as thus you can convert to your data to a normal geometry object instead. Use
the shape() function instead to convert a GeoJSON-like dict to a Shapely geometry.
• The empty() method on a geometry object is deprecated.
• The shapely.ops.cascaded_union function is deprecated. Use shapely.ops.unary_union instead, whichinternally already uses a cascaded union operation for better performance.

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CHAPTERTWO

INDICES AND TABLES• genindex• search

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INDEX

Symbols__eq__() (object method), 44Aalmost_equals() (object method), 45
area (object attribute), 28

Bboundary (object attribute), 48
bounds (object attribute), 28
buffer() (object method), 52
built-in function
prepared.prep(), 72
shapely.affinity.affine_transform(), 58
shapely.affinity.rotate(), 60
shapely.affinity.scale(), 61
shapely.affinity.skew(), 62
shapely.affinity.translate(), 63
shapely.geometry.asShape(), 78
shapely.geometry.box(), 35
shapely.geometry.mapping(), 78
shapely.geometry.polygon.orient(), 35
shapely.geometry.shape(), 78
shapely.ops.cascaded_union(), 67
shapely.ops.clip_by_rect(), 66
shapely.ops.linemerge(), 65
shapely.ops.nearest_points(), 70
shapely.ops.polygonize(), 65
shapely.ops.polygonize_full(), 65
shapely.ops.polylabel(), 74
shapely.ops.shared_paths(), 70
shapely.ops.snap(), 70
shapely.ops.split(), 71
shapely.ops.substring(), 71
shapely.ops.transform(), 63
shapely.ops.triangulate(), 68
shapely.ops.unary_union(), 66
shapely.ops.voronoi_diagram(), 69
shapely.wkb.dumps(), 76
shapely.wkb.loads(), 76
shapely.wkt.dumps(), 76
shapely.wkt.loads(), 76

validation.make_valid(), 73

C
centroid (object attribute), 48
contains() (object method), 45
convex_hull (object attribute), 54
covered_by() (object method), 45
covers() (object method), 45
crosses() (object method), 45

D
difference() (object method), 49
disjoint() (object method), 46
distance() (object method), 29

E
envelope (object attribute), 55
equals() (object method), 44

G
geom_type (object attribute), 29

H
has_z (object attribute), 42
hausdorff_distance() (object method), 29

I
interpolate() (object method), 41
intersection() (object method), 50
intersects() (object method), 46
is_ccw (object attribute), 42
is_empty (object attribute), 43
is_ring (object attribute), 43
is_simple (object attribute), 43
is_valid (object attribute), 43

Llength (object attribute), 28
LinearRing (built-in class), 32
LineString (built-in class), 30

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M
minimum_clearance (object attribute), 28
minimum_rotated_rectangle (object attribute), 55
MultiLineString (built-in class), 38
MultiPoint (built-in class), 37
MultiPolygon (built-in class), 39

N
nearest() (strtree.STRtree.strtree method), 75O
overlaps() (object method), 46

Pparallel_offset() (object method), 56
Point (built-in class), 30
Polygon (built-in class), 33
prepared.prep()built-in function, 72
project() (object method), 41

Q
query() (strtree.STRtree.strtree method), 75

Rrelate() (object method), 47
relate_pattern() (object method), 47
representative_point() (object method), 29

S
shapely.affinity.affine_transform()
built-in function, 58
shapely.affinity.rotate()
built-in function, 60
shapely.affinity.scale()built-in function, 61
shapely.affinity.skew()
built-in function, 62
shapely.affinity.translate()
built-in function, 63
shapely.geometry.asShape()
built-in function, 78
shapely.geometry.box()
built-in function, 35
shapely.geometry.CAP_STYLE (built-in variable), 52
shapely.geometry.JOIN_STYLE (built-in variable), 52
shapely.geometry.mapping()
built-in function, 78
shapely.geometry.polygon.orient()
built-in function, 35
shapely.geometry.shape()
built-in function, 78
shapely.ops.cascaded_union()90

built-in function, 67
shapely.ops.clip_by_rect()built-in function, 66
shapely.ops.linemerge()
built-in function, 65
shapely.ops.nearest_points()
built-in function, 70
shapely.ops.polygonize()built-in function, 65
shapely.ops.polygonize_full()built-in function, 65
shapely.ops.polylabel()built-in function, 74
shapely.ops.shared_paths()
built-in function, 70
shapely.ops.snap()
built-in function, 70
shapely.ops.split()
built-in function, 71
shapely.ops.substring()
built-in function, 71
shapely.ops.transform()
built-in function, 63
shapely.ops.triangulate()
built-in function, 68
shapely.ops.unary_union()built-in function, 66
shapely.ops.voronoi_diagram()
built-in function, 69
shapely.wkb.dumps()
built-in function, 76
shapely.wkb.loads()
built-in function, 76
shapely.wkt.dumps()
built-in function, 76
shapely.wkt.loads()
built-in function, 76
simplify() (object method), 58
strtree.STRtree (built-in class), 75
symmetric_difference() (object method), 50

T
touches() (object method), 46

Uunion() (object method), 50

Vvalidation.make_valid()
built-in function, 73

Wwithin() (object method), 46

Index