From: Hans Kristian Ruud <hkr@ifi.uio.no>
Subject: Point Location Summary - Another go
Organization: Dept. of Informatics, University of Oslo, Norway
Date: Wed, 18 Nov 1992 22:46:31 GMT
Originator: hkr@ull.ifi.uio.no

Hello.

I have been a bit slow in editing & reposting the messages I recieved
on point location - I'm sorry about that. 

Anyway, here goes:
 
 
-------------------------------------------------------------------------------

From pjyamamo@watdragon.uwaterloo.ca
 
I would suppose that a decision might be affected by the particular
application (type of polygons, type of queries, etc).

>Preparata & Shamos present several methods:
>
>  i)  The slab method.
> ii)  The chain method.
>iii)  The triangulation refinement method.
> iv)  The bridged chain method.
>
>The general agreement among the students was that method iii) seemed
>the best. I tend to agree, but I would like the opinions from the net
>on the matter.

In the same chapter they say that experimentally a bucketing technique
outperformed all others and that of the others, the trapezoid
decomposition performed best.

>A thousand thanks in advance (Paa forhaand tusen takk).

I would think that it also depends on how large the problem is.
Machine/System dependent problems may only start to show up in larger
problems. Ie because of the accesses to decompositions etc paging
could start to affect performance depending on implementation.

 
-------------------------------------------------------------------------------

From dougm@cs.rice.edu  (Doug Moore)

You might want to look at the article "Planar point location using
persistent search trees" by R. Tarjan and N. Sarnak in the
Communications of the ACM vol. 29 no 6 (1986).  It is theoretically
optimal, and seems rather simpler than triangulation refinement.

-------------------------------------------------------------------------------

From psu@cs.duke.edu 

Having taken a CG class taught from that book, and then worked in CG
for my thesis...

I would say that no one uses any of the methods in P&S in real life.  The
algorithms are simply too concerned with worst case asymptotic
optimality, and are thus overly complex, difficult to implement, and
probably slow.

I have heard that the trapezoid method is quick, but I've never seen
any code to prove it.

Also, one is hardly ever in the position of being able to sit and think
about how to pre-process the subdivision that we are locating
in...it's almost always an on-line process.

I have an incremental algorithm for constructing the Delaunay
triangulation (chap 5) that uses a bucketing scheme to do point
location.  This scheme will work much better than the general schemes
in Chap 2, but only on certain kinds of subdivisions (can't have too
many long edges, for example).  I can send you a TR that describes the
algorithm if you want.

My feeling is that no one PL scheme is best in general.  What you use
will depend on your application more than anything else.


-------------------------------------------------------------------------------


From moret@tiguex.cs.unm.edu (Bernard M.E. Moret)  


Much better (because much simpler and much faster) is the method of Sarnak
and Tarjan, using persistent data structures.  See their article in Commun. ACM
of 1986 (vol. 29, pp. 669-679).  It also generalizes easily to k-dimensions.
-- 
    Bernard M.E. Moret                       Department of Computer Science
    (505) 277-3112     University of New Mexico, Albuquerque, NM 87131-1386

-------------------------------------------------------------------------------

Compiled by Eric Haines, 3D/Eye Inc, 2359 Triphammer Rd, Ithaca, NY 14850
	erich@eye.com
All contents are copyright (c) 1992 by the individual authors
Archive locations: anonymous FTP at princeton.edu (128.112.128.1)
	/pub/Graphics/RTNews, wuarchive.wustl.edu:/graphics/graphics/RTNews,
	and many others.
UUCP archive access: write Kory Hamzeh (quad.com!avatar!kory) for info.

Contents:
    Introduction
    Intersection Between a Line and a Polygon (UNDECIDABLE??), by Dave Baraff,
	Tom Duff
    Fastest Point in Polygon Test, by Aladdin Nassar, Philip Walden, Eric
	Haines, Tom Dickens, Ron Capelli, Sundar Narasimhan, Christopher Jam,
	and (last but not least) Stuart MacMartin
    Polygon Intersection via Barycentric Coordinates, by Peter Shirley
    Many-Sided Polygon Intersection, by Eric Haines, Benjamin Zhu
    Code for Point in Polygon Intersectors, by Eric Haines

-------------------------------------------------------------------------------

Introduction

This special issue is dedicated to everyone's (well, at least my) favorite
computer graphics FAQ:  how do you find if a given point is inside a polygon?
John Griegg's FAQ posting points at _An Introduction to Ray Tracing_, edited
by Andrew Glassner.  This issue speeds up that algorithm (count the crossings
of a test ray), and hopefully lays to rest the idea of using the "sum the
angles" algorithm.  Plus there's a little discussion of what to do for special
cases such as triangles and complex polygons, and code for four different ways
to do this test at the end of the issue.

-------------------------------------------------------------------------------

[To begin this issue, here is one of the better pieces of obfuscatory writing
in the field of computer graphics.  Reprinted from RTNews8 - EAH]


Intersection Between a Line and a Polygon (UNDECIDABLE??),
	by Dave Baraff, Tom Duff

	From: deb@charisma.graphics.cornell.edu
	Newsgroups: comp.graphics
	Keywords: P, NP, Jordan curve separation, Ursyhon Metrization Theorem
	Organization: Program of Computer Graphics

In article [...] ncsmith@ndsuvax.UUCP (Timothy Lyle Smith) writes:
>
>  I need to find a formula/algorithm to determine if a line intersects
>  a polygon.  I would prefer a method that would do this in as little
>  time as possible.  I need this for use in a forward raytracing
>  program.

I think that this is a very difficult problem.  To start with, lines and
polygons are semi-algebraic sets which both contain uncountable number of
points.  Here are a few off-the-cuff ideas.

First, we need to check if the line and the polygon are separated.  Now, the
Jordan curve separation theorem says that the polygon divides the plane into
exactly two open (and thus non-compact) regions.  Thus, the line lies
completely inside the polygon, the line lies completely outside the polygon,
or possibly (but this will rarely happen) the line intersects the polyon.

Now, the phrasing of this question says "if a line intersects a polygon", so
this is a decision problem.  One possibility (the decision model approach) is
to reduce the question to some other (well known) problem Q, and then try to
solve Q.  An answer to Q gives an answer to the original decision problem.

In recent years, many geometric problems have been successfully modeled in a
new language called PostScript.  (See "PostScript Language", by Adobe Systems
Incorporated, ISBN # 0-201-10179-3, co. 1985).

So, given a line L and a polygon P, we can write a PostScript program that
draws the line L and the polygon P, and then "outputs" the answer.  By
"output", we mean the program executes a command called "showpage", which
actually prints a page of paper containing the line and the polygon.  A quick
examination of the paper provides an answer to the reduced problem Q, and thus
the original problem.

There are two small problems with this approach.

	(1) There is an infinite number of ways to encode L and P into the
	reduced problem Q.  So, we will be forced to invoke the Axiom of
	Choice (or equivalently, Zorn's Lemma).  But the use of the Axiom of
	Choice is not regarded in a very serious light these days.

	(2) More importantly, the question arises as to whether or not the
	PostScript program Q will actually output a piece of paper; or in
	other words, will it halt?

	Now, PostScript is expressive enough to encode everything that a
	Turing Machine might do; thus the halting problem (for PostScript) is
	undecidable.  It is quite possible that the original problem will turn
	out to be undecidable.


I won't even begin to go into other difficulties, such as aliasing, finite
precision and running out of ink, paper or both.

A couple of references might be:

1. Principia Mathematica.  Newton, I.  Cambridge University Press, Cambridge,
   England.  (Sorry, I don't have an ISBN# for this).

2. An Introduction to Automata Theory, Languages, and Computation.  Hopcroft, J
  and Ulman, J.

3. The C Programming Language. Kernighan, B and Ritchie, D.

4. A Tale of Two Cities. Dickens, C.

--------

From: td@alice.UUCP (Tom Duff)
Summary: Overkill.
Organization: AT&T Bell Laboratories, Murray Hill NJ

The situation is not nearly as bleak as Baraff suggests (he should know
better, he's hung around The Labs for long enough).  By the well known
Dobbin-Dullman reduction (see J. Dullman & D. Dobbin, J. Comp. Obfusc.
37,ii:  pp. 33-947, lemma 17(a)) line-polygon intersection can be reduced to
Hamiltonian Circuit, without(!) the use of Grobner bases, so LPI (to coin an
acronym) is probably only NP-complete.  Besides, Turing-completeness will no
longer be a problem once our Cray-3 is delivered, since it will be able to
complete an infinite loop in 4 milliseconds (with scatter-gather.)

--------

From: deb@svax.cs.cornell.edu (David Baraff)

Well, sure it's no worse than NP-complete, but that's ONLY if you restrict
yourself to the case where the line satisfies a Lipschitz condition on its
second derivative.  (I think there's an '89 SIGGRAPH paper from Caltech that
deals with this).

-------------------------------------------------------------------------------

Fastest Point in Polygon Test, by Aladdin Nassar, Philip Walden, Eric Haines,
	Tom Dickens, Ron Capelli, Sundar Narasimhan, Christopher Jam, and
	(last but not least) Stuart MacMartin


Aladdin Nassar asked a variant on the classic question:

What is the MOST EFFICIENT way to find if a point lies within a SIMPLE
polygon given a set of vertices (in no particular order) and the point
coordinate ?   Are there anonymous ftp sites with such canned functions
instead of re-inventing the wheel.  Excuse me if that is a trivial question.

----

Philip Walden (pwalden@hpcc01.corp.hp.com) replies:

Another alternative to using a test ray is to sum the arc angles from
the vertices to the test point. If the sum is 2pi, the point is inside,
if 0 it's outside. i.e.

	give a polygon with a set of vertices v[1:n] an test point p

	if (SUM i=1 to n (arcangle(v[i]->p->v[i+1])) > pi
	then p is inside
	else it's outside

----

Eric Haines replies:

This method turns out to be exceedingly slow, about an order of magnitude
slower than the "shoot a ray and count crossings" method.  The problem is that
the "arcangle" routine takes awhile:  you need to normalize your vectors, you
need to compute the dot product (for the angle) and cross product (for the
sign of the angle), and you need the arccosine of the dot product for the
angle itself.  This is done for every vertex of the polygon.  All of this adds
up!

In fact, I wrote a program to test the two algorithms head-to-head when I was
deciding which to use:  code attached (for a switch!).  The test program makes
polygons from 3 to 7 (though you can change this) and random scatters the
vertices from [0-1).  It then tests each of these polygons with a series of
random points from [0-1).

I won't claim that either implementation of the angle summation or segment
crossing routines below are optimal (though I do try to share intermediate
results between vertices).  But I don't care how much you tune the angle
summation algorithm:  it's still going to be slower than crossing counting.
Please do tell me of any hacks you make to get either algorithm faster,
anyway.

Code's at the end of this issue; you should mess with *_TEST_TIMES however you
see fit.  I found you could set them pretty low, but I felt better when they
were relatively high (i.e. the test times were greater than the granularity
of the "time" command).  The crossing counting inside/outside test is the one
I wrote up in _An Introduction to Ray Tracing_, edited by Andrew Glassner,
Academic Press, plus a trick or two not in the book.

I should also mention that the tests below assuming that an even winding
number means that the point is outside, as this is how the Jordan Curve test
is used.  Think of the winding number as the number of loops around a point.
As an example, think of a star shape:  the interior pentagon is considered
"outside" since points in it cross two lines.  You can modify the tests as
shown in the comments if you want to test for just a non-zero winding number
(i.e. the star is fully filled).

Timings are shown after the next comment.

----

Tom Dickens notes in a related article:

Additional logic [in the barycentric test] can be added to avoid checking
triangles which are located in a region which has already been determined to
be beyond of a side of a previous triangle which returned a negative check.

[I don't know if such an enhancement is possible (one triangle's outside is
another's inside) if the polygon is not convex, but mention it here for
interest's sake; it is certainly not incorporated in my code.  - EAH]

----

comments from Ron Capelli (capelli@vnet.ibm.com):

>the "arcangle" routine takes awhile:  you need to normalize your vectors, you
>need to compute the dot product (for the angle) and cross product (for the
>sign of the angle), and you need the arccosine of the dot product for the
>angle itself.  This is done for every vertex of the polygon.

See "Fast Approximation to the Arctangent" in Graphics Gems II (p.389).  The
maximum error of this approximation allows polygons with up to about 40
vertexes to be handled safely.

No need to normalize, take dot products, or cross products either.  From the
vector components (dx, dy) from a test point to a polygon vertex, get an angle
using arctan (or the fast approximation, in which case do not scale to radians
or degrees; use the angle value in the range 0 to 8).  Calculate the signed
angle between two successive vertexes by subtracting the angles for each
vertex.  Add up all the signed angle differences, and compare to the mid-range
value.

The case of a test point coincident with a polygon vertex must be explicitly
checked, as the arctangent for a zero-length vector is undefined.  Another
fine point is that angle differences must be wrapped in the range -180 to +180
degrees.

As fond as I am :-) of this method, I still won't argue that the "count ray
crossings" method is not better.  For a polygon with N edges, an optimized
"count ray crossings" algorithm requires at most one multiply, one divide, and
about a half-dozen add/subtract/compare operations per edge.  The "add angle
differences" algorithm using the fast approximation to the arctangent requires
only one divide but more than twice as many add/subtract/compare operations
per edge...

----

Sundar Narasimhan (sundar@fiber-one.ai.mit.edu) comments on Ron's post:

I think it is somewhat misleading to count worst case operations / edge and
then multiply it blindly by N esp.  since the "count ray crossings" algorithm
can be optimized to consider only the subset of N edges that actually have a
chance of intersecting the fired ray, whereas I haven't seen any such
pre-filtering done to the add angle difference algorithm.

----

Christopher Jam (phillips@swanee.ee.uwa.oz.au) writes:

For the "count ray crossings" method, if your processor has no multiply or
divide instructions, the multiply and divide may be replaced by a binary
search for the point on the edge with the same y coordinate as the point being
tested.  Alternatively, if you are going to draw the polygon at some stage,
generate a list of points on the line for plotting, and keep these in a table
to completely remove any requirements for multiplies, divides or searches.
Just look up a point on the same line ;)

----

Stuart MacMartin (sjm@bcrki65.bnr.ca) comments:

I know the purpose of your code was for a simple comparison of the two
algorithms, but for production you might want to write optimal code.

The loop overhead is actually quite expensive, as I found out when I optimized
someone else's implementation of this algorithm.  Most of the time is spent
looking for the next y that is on/below the ray or the next y that is above
the ray.  You should be able to get at lease a factor of 2 improvement by
optimizing those two loops.  Call a subroutine if necessary for the deleted
stuff.  (Note that subroutines can be MORE efficient because the compiler may
make a better use of registers for the time-consuming part of the loop.)  Use
pointers to the y rather than to the x to avoid the extra adding, and don't
keep track of so many variables in the loop.

Finally, is it necessary to use doubles?  With appropriate units, will long
suffice?  Just be careful if you make this switch:  if you have to have
doubles in some places in your code, watch out for conversions from long to
double and back.  On some platforms (definitely Apollo; I don't know about
others) such casts take so long that they totally swamp anything else that is
inefficient in a loop.

My code for the loop:

    stop = vertices + (numVertices << 1);

    for(y = vertices[1], p = vertices+3;  p < stop;  y = *p,p+=2) {
	/* Skip all lines above */
	if      (y > ay) while ((p < stop) && (*p > ay)) p += 2;
	/* Skip all lines below */
	else if (y < ay) while ((p < stop) && (*p < ay)) p += 2;
	    /* NOTE! Don't skip horizontal edges at ay! */
	if (p >= stop) break;

The routine requires that the first point is duplicated as the last.

----

Final (for now) comments, by Eric Haines:

Other people noted the barycentric test as being a way to do inside/outside
testing.  The barycentric test is included below in Peter Shirley's article,
and is also written up on page 390-393 of Graphics Gems, Didier Badouel's
article.  You can do better than figuring out all the intermediate differences
(as shown in Badouel's article), instead computing them as needed.  Also, my
code worries about zero area triangles just to be safe, so it could be a bit
faster without these sanity checks.

Badouel points out that triangle testing works for convex polygons:  you
simply test all of the triangles, and any intersection ends the test.  Most
people assume that concave triangles must be intersected using a different
test.  Not so!  As Berlin pointed out (Vol 11, SIGGRAPH '85 course notes -
sorry for the obscure reference!), you can test a concave polygon by checking
all its triangles generated from one vertex.  If the number of triangles hit
is odd, the point is inside the (potentially concave) polygon - try it out, it
actually works!  You do have to test all triangles, however; if you knew the
polygon was convex you can stop on the first intersection.  Also, the
barycentric coordinates become somewhat meaningless, since more than one
triangle can be found to overlap your test point.


So, included in the new code (which is at the end of this issue) are two
flavors of the crossings (segment) test, one which is fairly fast and
readable, the other is "macmartinized" by me and runs a bit faster still.
The barycentric polygon intersector is also included, and it works for all
polygons (concave, self-intersecting, etc).

Random polygon testing:

			 number (or range) of vertices
		    3	    4	    5	    3-7	    20	    100

angle		   57.21   69.81   86.44   82.98  303.33 1435.78
barycentric	    1.80    3.44    5.29    4.90   30.78  164.38
crossings	    2.17    2.76    3.32    3.24   12.51   60.13
macmartin	    1.86    2.35    2.87    2.64   11.04   52.27

inside %	    6.00   12.00   13.78    9.33   26.89   41.22

Times are in microseconds per intersection test, on an HP 720 workstation.
The "for" loop for repeating each test is counted into these test times
(basically, I repeat each polygon/point test a number of times so that there's
some reasonable amount of time to count) - not counting it just means the
"crossings" tests is all that much faster.  Basically, the macmartinized
crossing test wins on my machine; your mileage may vary.

Note that the barycentric intersector could be faster if the polygons are
known to be convex, since then a quick out could be taken (i.e. the first hit
ends testing).  I won't swear at all for the completeness of the barycentric
code:  I suspect you can get into roundoff error problems for intersection
points which lay exactly on some internal, invisible triangle testing edge
(rare, but possible).

There are also a few hacks I haven't done on the barycentric intersector:  you
could make the set of if statements into two big if statements, the variable
"v0" is not all that useful, etc.  Also, if the polygon is a triangle the test
could be made a special case and the "for" loop and other baggage could be
eliminated (this I tried, and it resulted in about a 20% reduction in run
time).  However, the code presented is at least readable, and the general
behavior is the same:  this intersector gets more expensive than the crossing
methods as the number of polygon vertices rises.

Looking over the branch flow analysis, I find that my test case polygons
(randomly generated points) cause a lot of algebraic tests to occur when using
the crossing algorithms (i.e.  the exact x intersection point has to be
computed when a polygon edge goes from one quadrant to its diagonal opposite).
For real polygons, this case is much less prevalent since vertices tend to be
close to each other, so the polygons I've generated are more pathological than
normally encountered.  For example, if the polygons generated are regular
polygons inscribed in a circle of radius 1.0 and given a random rotation about
the origin, the timings are

Regular polygon testing:

			 number (or range) of vertices
		    3	    4	    5	    3-7	    20	    100

angle		   54.20   70.80   85.14   87.83  314.53 1522.00
barycentric	    2.11    3.98    5.48    5.64   27.63  144.27
crossings	    2.50    3.09    3.30    3.42    9.46   42.71
macmartin	    2.32    2.81    2.91    2.92    6.29   23.91

inside %	   33.22   51.00   60.22   55.22   77.44   80.22

I'm not sure why there's any noticeable difference in the "angle summation"
tests, since this test should be the same for any test point (I suspect it's
just the "times()" routine's granularity).  The barycentric times change a
little for better and worse, probably due to times() and to different polygon
vs. point tests.  The two crossings test noticeably improve for large polygons
(29% and 54% savings, respectively) - this is due to less "difficult" edges,
and for the macmartin test there are now longer series of edges fully above or
fully below the test point.  These tests show off the real improvement of the
macmartin algorithm when testing many sided polygons.

The near-final word is that the crossings tests seems to be the most efficient
overall, the angle test should generally be avoided like the plague, and the
barycentric test is good for the special case of triangles (and you also
get the vertex interpolant weights).  Just to add to the confusion, I talk
about intersecting complex polygons in an article below; if you're using a lot
of these then another intersection algorithm might be a better route to go.

-------------------------------------------------------------------------------

Polygon Intersection via Barycentric Coordinates, by Peter Shirley
	(shirley@iuvax.cs.indiana.edu)

One good way to do ray triangle intersection is to use barycentric
coordinates.  If the triangle has vertices p0, p1, p2, then any point in the
plane containing the triangle can be represented as

	p = a*p0 + b*p2 + g*p3

and the constraint a + b + g = 1 will also hold.  If you want to interpolate
colors or normals across the triangle, you can use a similar formula.  A
little messing around will convince you that for points inside the triangle,
a, b, and g are all positive, and outside the triangle, at least one is
negative.  We can take this info, and get rid of a:

	p = p0 + b*(p1 - p0) + g*(p2-p0)

The point p is inside if b and g are positive, and (b+g) < 1.

A point on our ray can be represented as p' = o + t*v, where o is the point of
origin on the ray, and v is the direction of propagation.  Plugging this into
the triangle formula, we can find whether any point on the ray is also a point
on the plane:

	o + t*v = p0 + b*(p1 - p0) + g*(p2-p0)

Note that this is really 3 linear equations (one for x, y and z).  You can
rewrite this as a 3 by 3 linear system and solve for (t, b, g) and there is a
hit if and only if:

	t > 0
	b > 0
	g > 0
	b+g < 1

Alternatively, you can first find the equation of the plane containing the
triangle and find t.  This will give you an explicit point h on the plane.
You now have:

	h = p0 + b*(p1 - p0) + g*(p2-p0)

Now you have 3 equations and 2 unknowns.  Choosing any two of the equations is
equivalent to projecting the problem onto one of the cartesian planes.  Watch
out if the triangle is in one of the planes, because the projection might be a
line segment.  You can check the surface normal to see if it is one of those
cases (I just use the dimensions associated with the smallest magnitude
components of the normal).

Note-- none of this is new.  GG II and the ray tracing book have other ways to
do it.  Also, if your scenes are large, you are probably better off optimizing
your spatial partitioning code (the part which takes longer for bigger
scenes).

[For more info, see Didier Badouel's article in "Graphics Gems", p. 390-393]

-------------------------------------------------------------------------------

Many-Sided Polygon Intersection, by Eric Haines, Benjamin Zhu


Jon Bennett (jb7m+@andrew.cmu.edu) writes:

>For a project i was working on i needed a very fast 2-D point-in-polygon
>routine for a monte-carlo simulation, and came up with a variation of the
>standard jordan curve theorem algorithm which ran approximately on the
>order of O(n/20) when n > ~100 (for "mostly" round-ish polygons), instead
>of O(n).  What I can't find, and no one here seems to know, is for large
>polygons what is the best 2-D algorithm.  Everything they've heard of runs
>in O(n) (I know that n and n/20 are the same "order" but the you know what
>I mean). I'd like to know if there is something faster.


You might check Preparata & Shamos's _Computational Geometry_ book, I believe
they talk about faster algorithms.  Unfortunately, much of the stuff in CG
tends to be things like O(K * log n), where K is something pretty large.

A simple way to test polygons with a large number of edges in O(n/K) time is
to sort the edges into buckets, then find out which bucket the test point
falls into.  Then you do the usual ray test, but only against those edges in
the bucket.

For example, you could take a hundred edge polygon and find the Y extent in the
polygon's plane.  Split this extent into, say, 20 buckets (or whatever number
you like, more ==> more efficient, but also more memory and preprocessing).
Now take each edge and note which buckets it's in.  As an added speed-up, sort
the edges by their leftmost X values in that bucket.

Now testing can begin.  When you test a point, use the Y coordinate to find
which bucket to check.  Now test each edge by the normal Jordan test against
the point.  You only need to test edges to the left of the point (i.e. you are
using a test ray in the -X direction), so as soon as you reach an edge whose
leftmost X is greater than the point's X, you can stop testing.

The bucket sort immediately gets rid of a lot of edges, and the X test means
often not having to test all edges in the bucket (it's optional icing).

If you're really into using memory and like preprocessing, you could use a
grid structure to place buckets on the polygon.  Each grid cell is inside,
outside, or indeterminate.  There are some tricks, but I leave these as an
exercise for the reader...  Anyway, this method can be very fast, since much
of the polygon's area is classified as in or out and much of the time no edges
have to be tested at all.  The only limit is the grid cell resolution.  As the
resolution rises the actual testing approaches constant time ( O(1)!).
There's more preprocessing involved, but if you're making a lot of tests then
this is worth doing.

----

Benjamin Zhu (zhu@graphack.asd.sgi.com) comments on a similar question:

If your polygon is convex, (more generally, your polygon is star-shaped,) you
can pre-process the polygon into triangle strips by connecting any point in
the kernel of the polygon with each vertex of the polygon.  This takes you
O(n) pre-processing time.  After that, for each point, you do a binary search
in polar coordinates to locate the potential triangle where the point might
lie.  Then you can determine if the point lies within this triangle, which is
trivial.  See Preparata and Shamos' "Computational Geometry" for more details.
Obviously, this algorithm will give you O(log(n)) time per point.  So you
might consider this one if you need to solve the point-inclusion problem many
times.

-------------------------------------------------------------------------------

Code for Point in Polygon Intersectors, by Eric Haines

Here it is, code for the angle test, the barycentric test, and the two
crossings tests, with a main program to test their speeds.  You may need to
use something other than srand()/drand48() for your random number generator,
and the times() command for timing.  You'll also want to change the
*_TEST_TIMES constants if you're using something slower than an HP 720
workstation.  Feel free to complain about the slowness of any of the code -
I'm always interested in new hacks.


/* Point in polygon inside/outside tester.  Angle summation, barycentric
 * coordinates, and ray along x-axis (crossings testing) compared.
 *
 * copyright 1992 by Eric Haines, 3D/Eye Inc, erich@eye.com
 */

#include <math.h>
#include <sys/types.h>
#include <sys/param.h>
#include <sys/times.h>

#define	X	0
#define	Y	1

double	drand48() ;

double	AngleTimeTotal ;
double	BarycentricTimeTotal ;
double	CrossingsTimeTotal ;
double	MacmartinTimeTotal ;

/* minimum & maximum number of polygon vertices to generate */
#define	MIN_VERTS	3
#define	MAX_VERTS	7

/* number of different polygons to try */
#define	TEST_POLYGONS	30

/* number of different intersection points to try */
#define	TEST_POINTS	30

/* number of times to try a single point vs. a polygon */
/* this should be > 1 / ( HZ * approx. single test time in seconds ) */
#define	ANGLE_TEST_TIMES	1000
#define	BARYCENTRIC_TEST_TIMES	10000
#define	CROSSINGS_TEST_TIMES	10000
#define	MACMARTIN_TEST_TIMES	10000

main(argc,argv)
int argc;  char *argv[];
{
int	i, j, numverts, inside_tot ;
int	angle_flag, barycentric_flag, crossings_flag, macmartin_flag ;
double	pgon[MAX_VERTS][2] ;
double	point[2] ;

    srand( 12345 ) ;

    AngleTimeTotal = 0.0 ;
    BarycentricTimeTotal = 0.0 ;
    CrossingsTimeTotal = 0.0 ;
    MacmartinTimeTotal = 0.0 ;
    inside_tot = 0 ;

    for ( i = 0 ; i < TEST_POLYGONS ; i++ ) {

#ifdef CENTERED_SQUARE
	/* for debugging purposes, test against a square */
	numverts = 4 ;
	pgon[0][X] = 0.2 ;
	pgon[0][Y] = 0.2 ;
	pgon[1][X] = 0.8 ;
	pgon[1][Y] = 0.2 ;
	pgon[2][X] = 0.8 ;
	pgon[2][Y] = 0.8 ;
	pgon[3][X] = 0.2 ;
	pgon[3][Y] = 0.8 ;
#else
	/* make an arbitrary polygon fitting 0-1 range in x and y */
	numverts = MIN_VERTS +
		(int)(drand48() * (double)(MAX_VERTS-MIN_VERTS+1)) ;
	for ( j = 0 ; j < numverts ; j++ ) {
	    pgon[j][X] = drand48() ;
	    pgon[j][Y] = drand48() ;
	}
#endif

	/* now try # of points against it */
	for ( j = 0 ; j < TEST_POINTS ; j++ ) {
	    point[X] = drand48() ;
	    point[Y] = drand48() ;
	    angle_flag = angletest( pgon, numverts, point ) ;
	    barycentric_flag = barycentrictest( pgon, numverts, point ) ;
	    crossings_flag = crossingstest( pgon, numverts, point ) ;
	    macmartin_flag = macmartintest( pgon, numverts, point ) ;

	    /* reality check */
	    if ( angle_flag != crossings_flag ) {
		printf( "angle test says %s, crossings test says %s\n",
		    angle_flag ? "INSIDE" : "OUTSIDE",
		    crossings_flag ? "INSIDE" : "OUTSIDE" ) ;
		printf( "point %g %g\n", (float)point[X], (float)point[Y] ) ;
		printf( "polygon:\n" ) ;
		for ( j = 0 ; j < numverts ; j++ ) {
		    printf( "  %g %g\n", (float)pgon[j][X], (float)pgon[j][Y]);
		}
	    }
	    if ( barycentric_flag != macmartin_flag ) {
		printf( "barycentric test says %s, macmartin test says %s\n",
		    barycentric_flag ? "INSIDE" : "OUTSIDE",
		    macmartin_flag ? "INSIDE" : "OUTSIDE" ) ;
		printf( "point %g %g\n", (float)point[X], (float)point[Y] ) ;
		printf( "polygon:\n" ) ;
		for ( j = 0 ; j < numverts ; j++ ) {
		    printf( "  %g %g\n", (float)pgon[j][X], (float)pgon[j][Y]);
		}
	    }

	    inside_tot += crossings_flag ;
	}
    }
    printf( "angle test time: %g microseconds per test\n",
	(float)( AngleTimeTotal/(double)(TEST_POINTS*TEST_POLYGONS) ) ) ;
    printf( "barycentric test time: %g microseconds per test\n",
	(float)( BarycentricTimeTotal/(double)(TEST_POINTS*TEST_POLYGONS) ) ) ;
    printf( "crossings test time: %g microseconds per test\n",
	(float)( CrossingsTimeTotal/(double)(TEST_POINTS*TEST_POLYGONS) ) ) ;
    printf( "macmartin crossings test time: %g microseconds per test\n",
	(float)( MacmartinTimeTotal/(double)(TEST_POINTS*TEST_POLYGONS) ) ) ;

    printf( "%g %% of all points were inside polygons\n",
	(float)inside_tot * 100.0 / (float)(TEST_POINTS*TEST_POLYGONS) ) ;
}

/* sum angles of vtxN - point - vtxN+1, check if in PI to 3*PI range */
int
angletest( pgon, numverts, point )
double	pgon[MAX_VERTS][2] ;
int	numverts ;
double	point[2] ;
{
int	i, j, inside_flag ;
struct  tms     timebuf ;
long	timestart ;
long	timestop ;
double	*vtx0, *vtx1, angle, len, vec0[2], vec1[2] ;

    /* do the test a bunch of times to get a useful time reading */
    timestart = times( &timebuf ) ;
    for ( i = 0 ; i < ANGLE_TEST_TIMES ; i++ ) {
	/* sum the angles and see if answer mod 2*PI > PI */
	vtx0 = pgon[numverts-1] ;
	vec0[X] = vtx0[X] - point[X] ;
	vec0[Y] = vtx0[Y] - point[Y] ;
	if ( (len = hypot( vec0[X], vec0[Y] )) <= 0.0 ) {
	    /* point and vertex coincide */
	    return( 1 ) ;
	}
	vec0[X] /= len ;
	vec0[Y] /= len ;

	angle = 0.0 ;
	for ( j = 0 ; j < numverts ; j++ ) {
	    vtx1 = pgon[j] ;
	    vec1[X] = vtx1[X] - point[X] ;
	    vec1[Y] = vtx1[Y] - point[Y] ;
	    if ( (len = hypot( vec1[X], vec1[Y] )) <= 0.0 ) {
		/* point and vertex coincide */
		return( 1 ) ;
	    }
	    vec1[X] /= len ;
	    vec1[Y] /= len ;

	    /* check if vec1 is to "left" or "right" of vec0 */
	    if ( vec0[X] * vec1[Y] - vec1[X] * vec0[Y] >= 0.0 ) {
		/* add angle due to dot product of vectors */
		angle += acos( vec0[X] * vec1[X] + vec0[Y] * vec1[Y] ) ;
	    } else {
		/* subtract angle due to dot product of vectors */
		angle -= acos( vec0[X] * vec1[X] + vec0[Y] * vec1[Y] ) ;
	    }

	    /* get to next point */
	    vtx0 = vtx1 ;
	    vec0[X] = vec1[X] ;
	    vec0[Y] = vec1[Y] ;
	}
	/* test if between PI and 3*PI, 5*PI and 7*PI, etc */
	/* if we care about is winding number > 0, then just:
	       inside_flag = fabs(angle) > M_PI ;
	 */
	inside_flag = fmod( fabs(angle) + M_PI, 4.0*M_PI ) > 2.0*M_PI ;
    }
    timestop = times( &timebuf ) ;
    /* time in microseconds */
    AngleTimeTotal += 1000000.0 * (double)(timestop - timestart) /
	(double)(HZ * ANGLE_TEST_TIMES) ;

    return (inside_flag) ;
}

/* barycentric, a la Gems I, with a little efficiency tuning */
int
barycentrictest( pgon, numverts, point )
double	pgon[MAX_VERTS][2] ;
int	numverts ;
double	point[2] ;
{
int	i, tris_hit, inside_flag, p1, p2 ;
struct  tms     timebuf ;
long	timestart ;
long	timestop ;
double	tx, ty, u0, u1, u2, v0, v1, alpha, beta, denom ;

    /* do the test a bunch of times to get a useful time reading */
    timestart = times( &timebuf ) ;
    for ( i = 0 ; i < BARYCENTRIC_TEST_TIMES ; i++ ) {

	tx = point[X] ;
	ty = point[Y] ;

	tris_hit = 0 ;

	for ( p1 = 1, p2 = 2 ; p2 < numverts ; p1++, p2++ ) {

	    if ( ( u1 = pgon[0][X] - pgon[p2][X] ) == 0.0 ) {

		/* zero area test - optional */
		if ( ( u2 = pgon[p1][X] - pgon[p2][X] ) == 0.0 ) {
		    goto NextTri;
		}

		/* Compute intersection point */
		if ( ( ( beta = ( tx - pgon[p2][X] ) / u2 ) < 0.0 ) ||
		       ( beta > 1.0 ) ) {

		    goto NextTri;
		}

		if ( ( v1 = pgon[0][Y] - pgon[p2][Y] ) == 0.0 ) {
		    goto NextTri;
		}

		if ( ( alpha = ( ty - pgon[p2][Y] - beta *
			( pgon[p1][Y] - pgon[p2][Y] ) ) / v1 ) < 0.0 ) {
		    goto NextTri;
		}

	    } else {
		if ( ( denom = ( pgon[p1][Y] - pgon[p2][Y] ) * u1 -
			( u2 = pgon[p1][X] - pgon[p2][X] ) *
			( v1 = pgon[0][Y] - pgon[p2][Y] ) ) == 0.0 ) {

		    goto NextTri;
		}

		/* Compute intersection point & subtract 0 vertex */
		u0 = tx - pgon[p2][X] ;
		v0 = ty - pgon[p2][Y] ;

		if ( ( ( ( beta = ( v0 * u1 - u0 * v1 ) / denom ) ) < 0.0 ) ||
		       ( beta > 1.0 ) ) {

		    goto NextTri;
		}
		if ( ( alpha = ( u0 - beta * u2 ) / u1 ) < 0.0 ) {
		    goto NextTri;
		}
	    }

	    /* check gamma */
	    if ( alpha + beta <= 1.0 ) {
		/* survived */
		tris_hit++ ;
	    }

	    NextTri: ;
	}
	inside_flag = tris_hit & 0x1 ;
    }
    timestop = times( &timebuf ) ;
    /* time in microseconds */
    BarycentricTimeTotal += 1000000.0 * (double)(timestop - timestart) /
	(double)(HZ * BARYCENTRIC_TEST_TIMES) ;

    return (inside_flag) ;
}

/* shoot a test ray along +X axis - slower version, less messy */
int
crossingstest( pgon, numverts, point )
double	pgon[MAX_VERTS][2] ;
int	numverts ;
double	point[2] ;
{
int	i, j, inside_flag, xflag0 ;
struct  tms     timebuf ;
long	timestart ;
long	timestop ;
double	*vtx0, *vtx1, dv0 ;
int	crossings, yflag0, yflag1 ;

    /* do the test a bunch of times to get a useful time reading */
    timestart = times( &timebuf ) ;
    for ( i = 0 ; i < CROSSINGS_TEST_TIMES ; i++ ) {

	vtx0 = pgon[numverts-1] ;
	/* get test bit for above/below Y axis */
	yflag0 = ( dv0 = vtx0[Y] - point[Y] ) >= 0.0 ;

	crossings = 0 ;
	for ( j = 0 ; j < numverts ; j++ ) {
	    /* cleverness:  bobble between filling endpoints of edges, so
	     * that the previous edge's shared endpoint is maintained.
	     */
	    if ( j & 0x1 ) {
		vtx0 = pgon[j] ;
		yflag0 = ( dv0 = vtx0[Y] - point[Y] ) >= 0.0 ;
	    } else {
		vtx1 = pgon[j] ;
		yflag1 = ( vtx1[Y] >= point[Y] ) ;
	    }

	    /* check if points not both above/below X axis - can't hit ray */
	    if ( yflag0 != yflag1 ) {
		/* check if points on same side of Y axis */
		if ( ( xflag0 = ( vtx0[X] >= point[X] ) ) ==
		     ( vtx1[X] >= point[X] ) ) {

		    if ( xflag0 ) crossings++ ;
		} else {
		    /* compute intersection of pgon segment with X ray, note
		     * if > point's X.
		     */
		    crossings += (vtx0[X] -
			dv0*( vtx1[X]-vtx0[X])/(vtx1[Y]-vtx0[Y])) >= point[X] ;
		}
	    }
	}
	/* test if crossings is odd */
	/* if we care about is winding number > 0, then just:
	       inside_flag = crossings > 0 ;
	 */
	inside_flag = crossings & 0x01 ;
    }
    timestop = times( &timebuf ) ;
    /* time in microseconds */
    CrossingsTimeTotal += 1000000.0 * (double)(timestop - timestart) /
	(double)(HZ * CROSSINGS_TEST_TIMES) ;

    return (inside_flag) ;
}

/* shoot a test ray along +X axis - macmartinized by me, a bit messier */
int
macmartintest( pgon, numverts, point )
double	pgon[MAX_VERTS][2] ;
int	numverts ;
double	point[2] ;
{
int	i, inside_flag, xflag0 ;
struct  tms     timebuf ;
long	timestart ;
long	timestop ;
double	*p, *stop ;
int	crossings ;
double	tx, ty, y ;

    /* do the test a bunch of times to get a useful time reading */
    timestart = times( &timebuf ) ;
    for ( i = 0 ; i < MACMARTIN_TEST_TIMES ; i++ ) {

	crossings = 0 ;

	tx = point[X] ;
	ty = point[Y] ;
	y = pgon[numverts-1][Y] ;

	p = (double *)pgon + 1 ;
	if ( ( y >= ty ) != ( *p >= ty ) ) {
	    /* x crossing */
	    if ( ( xflag0 = ( pgon[numverts-1][X] >= tx ) ) ==
		 ( *(double *)pgon >= tx ) ) {

		if ( xflag0 ) crossings++ ;
	    } else {
		/* compute intersection of pgon segment with X ray, note
		 * if > point's X.
		 */
		crossings += ( pgon[numverts-1][X] -
		(y-ty)*( *(double *)pgon - pgon[numverts-1][X])/(*p-y)) >= tx ;
	    }
	}

	stop = pgon[numverts] ;

	for ( y=*p,p += 2 ; p < stop ; y=*p,p+=2) {

	    if ( y >= ty ) {
		while ( (p < stop) && (*p >= ty) ) p+=2 ;
		if ( p >= stop ) goto Exit ;
		/* y crosses */
		if ( ( xflag0 = ( *(p-3) >= tx ) ) ==
		     ( *(p-1) >= tx ) ) {

		    if ( xflag0 ) crossings++ ;
		} else {
		    /* compute intersection of pgon segment with X ray, note
		     * if > point's X.
		     */
		    crossings += ( *(p-3) -
			(*(p-2)-ty)*( *(p-1)-*(p-3))/(*p-*(p-2))) >= tx ;
		}
	    } else {
		while ( (p < stop) && (*p < ty)) p+=2 ;
		if ( p >= stop ) goto Exit ;
		/* y crosses */
		if ( ( xflag0 = ( *(p-3) >= tx ) ) ==
		     ( *(p-1) >= tx ) ) {

		    if ( xflag0 ) crossings++ ;
		} else {
		    /* compute intersection of pgon segment with X ray, note
		     * if > point's X.
		     */
		    crossings += ( *(p-3) -
			(*(p-2)-ty)*( *(p-1)-*(p-3))/(*p-*(p-2))) >= tx ;
		}
	    }
	}

	Exit:
	/* test if crossings is odd */
	/* if we care about is winding number > 0, then just:
	       inside_flag = crossings > 0 ;
	 */
	inside_flag = crossings & 0x01 ;
    }
    timestop = times( &timebuf ) ;
    /* time in microseconds */
    MacmartinTimeTotal += 1000000.0 * (double)(timestop - timestart) /
	(double)(HZ * MACMARTIN_TEST_TIMES) ;

    return (inside_flag) ;
}
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