Approaches - Revision history

Arnaud Cheritat: /* Backward and forward iteration */

2016-07-28T07:58:42Z

‎Backward and forward iteration

2016-02-23T13:50:14Z

‎Scanline and direct

2015-12-20T19:35:52Z

‎Example of scanline algorithm with forward iteration

2015-12-18T22:04:37Z

‎Image/preimage of a set

2014-04-20T20:48:46Z

‎Scanline and direct

2014-04-20T20:26:57Z

‎Image/preimage of a set

2014-04-20T15:40:46Z

‎Scanline and direct

2014-04-20T15:02:56Z

2014-04-20T15:01:41Z

‎Example of an inverse iteration method

2014-04-20T14:11:28Z

‎Backward and forward iteration

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 Forward iteration means that we compute z<sub>1</sub>, z<sub>2</sub>=f(z<sub>1</sub>), z<sub>3</sub>=f(z<sub>2</sub>), etc...
-Backward iteration means that $z_2=f^{-1}(z_1)$, $z_3=f^{-1}(z_2)$, etc... The solution may be non-unique.
+Backward iteration means that $z_2=f^{-1}(z_1)$, $z_3=f^{-1}(z_2)$, etc... The solution $z$ of $f(z)=w$ may be non-unique.
 Most of my scanline programs use forward iteration.

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 When creating an image there are two main methods:
-* scanline methods: all pixels in the image will be scanned. For each pixel, the color has to be determined. The coordinates of the pixel are converted into mathematical parameters. Then an algorithm is run on that parameter. For instance we may have a dynamical system, the point may represent a point in phase space, whose orbit we simulate/iterate, and the color of the pixel will be chosen according to the behaviour of the orbit.
+* scanline methods: (a.k.a. raster scan) all pixels in the image will be scanned. For each pixel, the color has to be determined. The coordinates of the pixel are converted into mathematical parameters. Then an algorithm is run on that parameter. For instance we may have a dynamical system, the point may represent a point in phase space, whose orbit we simulate/iterate, and the color of the pixel will be chosen according to the behaviour of the orbit.
 * direct drawing: points, lines or areas, are determined by their mathematical coordinates, converted into pixel coordinates, and drawn.

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 **  Set the complex number z=0
 **  Do the following at most N times:
-***     if the module of z is bigger that 4:
+***     if the modulus of z is bigger that 2:
 ****         color the pixel (i,j) white
 ****         exit the loop and pass to the next pixel

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 Example: you want to compute the n-th preimage by f of the upper half plane. You may for each pixel compute the corresponding point z, then test wether or not f<sup>n</sup>(z) has positive imaginary part and color the pixel white or black according to the result.
-Example: you want to compute the n-th image by f of a circle. You may discretize the circle into certain number of points z, compute f<sup>n</sup>(z) for each of them, and plot a dot at each resulting value, or a straigt line between successive values. There are more clever ways to do this, though: for instance by adapting the step to the movement of the see point [[section to be added]]. One could also use higher order approximations, like Bezier curves, but this becomes complicated.
+Example: you want to compute the n-th image by f of a circle. You may discretize the circle into certain number of points z, compute f<sup>n</sup>(z) for each of them, and plot a dot at each resulting value, or a straigt line between successive values. There are more clever ways to do this, though: for instance by adapting the step to the movement of the point, see [[section to be added]]. One could also use higher order approximations, like Bezier curves, but this becomes complicated.
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 * direct drawing: points, lines or areas, are determined by their mathematical coordinates, converted into pixel coordinates, and drawn.
-Direct drawing is a priori much faster as, in good cases, it focuses targets computation only at necessary pixels.
+Direct drawing is a priori much faster as, in good cases, it targets computation only at necessary pixels.
 Direct drawing is also the only approach compatible with vector graphics.

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 * In scanline methods, one computes the n-th inverse image of a set
 * In direct methods, one computes the n-th image of a set
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 ** let z have the value w and repeat the process
-It has defects: it draws the same pixels many times and takes a lot of time to find some other pixels. This is well explained in the book ''The beauty of Fractals''<ref>Heinz-Otto Peitgen, Peter Richter, ''The beauty of fractals'', Springer-Verlag, Heidelberg, 1986, 0-387-15851-0</ref>.
+It has defects: it draws the same pixels many times and takes a lot of time to find some other pixels. This is well explained in the book ''The beauty of Fractals''<ref>Heinz-Otto Peitgen, Peter Richter, ''The beauty of fractals'', Springer-Verlag, Heidelberg, 1986, ISBN 0-387-15851-0</ref>.
 == Image/preimage of a set ==
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 Backward iteration means that $z_2=f^{-1}(z_1)$, $z_3=f^{-1}(z_2)$, etc... The solution may be non-unique.
-A vast majority of my scanline programs use forward iteration.
+Most of my scanline programs use forward iteration.
 === Example of scanline algorithm with forward iteration ===
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 There are better algorithms, see the special page [[Mandelbrot set]] on this wiki.
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