affine transformations The most frequently used transformation has A 85 04 04

Affine transformations the most frequently used

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affine transformations . The most frequently used transformation has A = 0 . 85 0 . 04 - 0 . 04 0 . 85 , b = 0 1 . 6 . This transformation shortens and rotates x a little bit, then adds 1 . 6 to its second component. Repeated application of this transformation moves the point up and to the right, heading toward the upper tip of the fern. Every once in a while, one of the other three transformations is picked at random. These transformations move the point into the lower subfern on the right, the lower subfern on the left, or the stem. Here is the complete fractal fern program. function fern %FERN MATLAB implementation of the Fractal Fern %Michael Barnsley, Fractals Everywhere, Academic Press,1993 %This version runs forever, or until stop is toggled. %See also: FINITEFERN. shg clf reset set(gcf,’color’,’white’,’menubar’,’none’, ... ’numbertitle’,’off’,’name’,’Fractal Fern’) x = [.5; .5]; h = plot(x(1),x(2),’.’); darkgreen = [0 2/3 0]; set(h,’markersize’,1,’color’,darkgreen,’erasemode’,’none’); axis([-3 3 0 10]) axis off stop = uicontrol(’style’,’toggle’,’string’,’stop’, ... ’background’,’white’); drawnow p = [ .85 .92 .99 1.00]; A1 = [ .85 .04; -.04 .85]; b1 = [0; 1.6]; A2 = [ .20 -.26; .23 .22]; b2 = [0; 1.6]; A3 = [-.15 .28; .26 .24]; b3 = [0; .44]; A4 = [ 0 0 ; 0 .16]; cnt = 1;
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16 Chapter 1. Introduction to MATLAB tic while ~get(stop,’value’) r = rand; if r < p(1) x = A1*x + b1; elseif r < p(2) x = A2*x + b2; elseif r < p(3) x = A3*x + b3; else x = A4*x; end set(h,’xdata’,x(1),’ydata’,x(2)); cnt = cnt + 1; drawnow end t = toc; s = sprintf(’%8.0f points in %6.3f seconds’,cnt,t); text(-1.5,-0.5,s,’fontweight’,’bold’); set(stop,’style’,’pushbutton’,’string’,’close’, ... ’callback’,’close(gcf)’) Let’s examine this program a few statements at a time. shg stands for “show graph window.” It brings an existing graphics window forward, or creates a new one if necessary. clf reset resets most of the figure properties to their default values. set(gcf,’color’,’white’,’menubar’,’none’, ... ’numbertitle’,’off’,’name’,’Fractal Fern’) changes the background color of the figure window from the default gray to white and provides a customized title for the window. x = [.5; .5]; provides the initial coordinates of the point. h = plot(x(1),x(2),’.’); plots a single dot in the plane and saves a handle , h , so that we can later modify the properties of the plot. darkgreen = [0 2/3 0];
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1.3. Fractal Fern 17 defines a color where the red and blue components are zero and the green component is two-thirds of its full intensity. set(h,’markersize’,1,’color’,darkgreen,’erasemode’,’none’); makes the dot referenced by h smaller, changes its color, and specifies that the image of the dot on the screen should not be erased when its coordinates are changed. A record of these old points is kept by the computer’s graphics hardware (until the figure is reset), but Matlab itself does not remember them.
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