# Fractals/Computer graphic techniques/2D/plane

2D plane direct links:
* coordinate
* transformations ( maps )
* quality of the image
* graphic files
* parameter files
* Size and resolution of the image
* Scanning, sampling and decomposition

# Geometry

* Euclidean
* non-Euclidean
** hyperbolic
** eliptic
Quality of image geometry:
* good ( not distorted)
* bad ( distorted)

# visualisation

* grid
* level sets
** Zebra striping in computer graphics
* checker board ( chess board)

# Viewport (visible part of the plane )

A viewport is a polygon viewing region in computer graphics.

## Description

Rectangle part of 2D plane can be described by :
* corners ( 4 real numbers or 2 complex numbers = 2D points)
* center and :
** width ( 3 real numbers )
** magnification
** radius
"People specify fractal coordinates in many ways. Some people use the coordinates of the upper-left and lower-right visible points, specifying the coordinates as four numbers x1, y1, x2, y2. To set the same viewpoint in XaoS, set the real portion of the center to (x1+x2)/2, the imaginary part of center to (y1+y2)/2, and the radius to the greater of x2-x1 and y2-y1." ( from Xaos doc )

### Corners

Standard description in Fractint, Ultra Fractal, ChaosPro and Fractal Explorer are corners. For example initial plane for Mandelbrot set is
Display window of parameter plane has :
* a horizontal width of 4 (real)
* a vertical width (height) of 3 (imag)
* an aspect ratio (proportion) 4/3 ( also in pixels 640/480 so ther is no distorsion)
* center z=-0.5
See demo par file :
For julia set/ dynamic plane has :
Description from documentation of Fractint :
CORNERS=[xmin/xmax/ymin/ymax[/x3rd/y3rd]]
Example: corners=-0.739/-0.736/0.288/0.291
Begin with these coordinates as the range of x and y coordinates, rather than the default values of (for type=mandel) -2.0/2.0/-1.5/1.5. When you specify four values (the usual case), this defines a rectangle: x- coordinates are mapped to the screen, left to right, from xmin to xmax, y-coordinates are mapped to the screen, bottom to top, from ymin to ymax. Six parameters can be used to describe any rotated or stretched parallelogram: (xmin,ymax) are the coordinates used for the top-left corner of the screen, (xmax,ymin) for the bottom-right corner, and (x3rd,y3rd) for the bottom-left. Entering just "CORNERS=" tells Fractint to use this form (the default mode) rather than CENTER-MAG (see below) when saving parameters with the [B] command.

### Center and ...
#### magnification

Fractint uses Mag ( compare with Xmagfactor)
This is an alternative way to enter corners as a center point and a magnification that is popular with some fractal programs and publications. Entering just "CENTER-MAG=" tells Fractint to use this form rather than CORNERS (see above) when saving parameters with the [B] command. The [TAB] status display shows the "corners" in both forms. When you specify three values (the usual case), this defines a rectangle: (Xctr, Yctr) specifies the coordinates of the center of the image.
Mag indicates the amount of magnification to use. Initial value ( no zoom ) is 6.66666667e-01.
Six parameters can be used to describe any rotated or stretched parallelogram: Xmagfactor tells how many times bigger the x- magnification is than the y-magnification,
Rotation indicates how many degrees the image has been turned.
Skew tells how many degrees the image is leaning over. Positive angles will rotate and skew the image counter-clockwise.
Parameters can be saved to parmfile called fractint.par

#### width

Wolf Jung uses center and width in Mandel:

#### width and height

to describe plane ( view ) Xaos uses:
* 4 numbers in scripts
* 3 numbers in menu
Xaos uses to call it "radius" but it is defind as : the greater of (x2-x1= width) and y2-y1=height."

#### radius

It is very usefull for zooming : fix center and only change radius.
Claude Heiland-Allen use center and radius for parameter plane of complex quadratic polynomial:
Radius is defined as:
* the radius of a circle that fits within the actual frame of the image
* "the difference in imaginary coordinate between the center and the top of the axis-aligned view rectangle".
* "px_radius is half the width of the image (in real coordinates)"
$radius={\frac {1}{Mag}}={\frac {height}{2}}$
another version
The same in GLSL ( for Shadertoy) :
Set plane corners from radius, center and aspect ratio:

# Orientation

There is a major incompatibility between:
* the standard Math convention taught in Math class = standard coordinates (y-up)
* some CS graphics programming systems (but not all) = y-down
Check orientation of the plane by marking first quadrant of Cartesian plane :
It should be in upper right position.
See also
* math.stackexchange question: proper-name-for-inverted-cartesian-coordinate-system

# Aspect ratio

* image
* display

## image

The aspect ratio of an image is the ratio of its width to its height, and is expressed with two numbers separated by a colon, such as 16:9, sixteen-to-nine. For the x:y aspect ratio, the image is x units wide and y units high. Common aspect ratios are 1.85:1 and 2.39:1 in cinematography, 4:3 and 16:9 in television photography, and 3:2 in still photography.

## Display Aspect ratio

The aspect ratio, or more precisely the Display Aspect Ratio (DAR) – the aspect ratio of the image as displayed. For TV:
* for TV DAR as traditionally 4:3 (a.k.a. Fullscreen)
* now is 16:9 (a.k.a. Widescreen) now the standard for HDTV

# Optimisation

* "recursive algorithm to split the image up into blocks, and tests each block to see whether it lies inside a “black area”." by Michael Hogg
* Poincaré half-plane metric for zoom animation
* A Simple Optimization of Fractal Animation by Wei-Yin Chen, You-Sheng Yang and Kun-Mao Liang
* optimizing zoom animations by Claude Heiland-Allen
* reenigne blog : recursive subdivision
* Xaos - algorithm description
* New fast method !!!!
* Zooming by Albert Lobo Cusidó
* Rendering the Mandelbrot Set — with an example implementation in javascript By Christian Stigen Larsen
* perturbation

# Patterns

* Stripe Patterns on Surfaces" given by Keenan Crane

# References

* ↑ Euclidean space in wikipedia
* ↑ Non-Euclidean Geometry by Malin Christersson ( 2018)
* ↑ viewport in wikipedia
* ↑ Xaos - view
* ↑ Claude Heiland-Allen blog
* ↑ Exponential Map by Robert P. Munafo, 2010 Dec 5. From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020.
* ↑ stackoverflow question: is-opengl-coordinate-system-left-handed-or-right-handed
* ↑ wikipedia : Aspect_ratio_(image)
* ↑ michael hogg : fractalnet
* ↑ optimizing zoom animations by Claude Heiland-Allen