Animation of Conformal Mappings - matlab

Question

In this movie the conformal mapping fz = -1/z is visualized as a continuous transformation of an image to the distorted result. How was this made? I am working with conformal mappings in Matlab and I would like to know a method to generate the intermediate steps of an arbitrary conformal transformation in analogy to this movie.

Answer 1

One old trick is to use surface with FaceColor = texturemap with Z=0. A texturemapped surface with Z=0 is basically an image, but it lets you mess with the X&Y coordinates of the underlying grid to do this sort of warping.
You'd want 2D arrays for your XData & YData. Those are the ones you'll manipulate with the mapping. Your ZData would be an array of the same size which was all zeros. Then the CData array would be an image. If you wanted to use a 20x20 grid, it might look something like this:
m = 20;
n = 20;
[x,y] = meshgrid(m,n);
img = imread('myimage.png');
h = surf(x,y,zeros(m,n),img,'FaceColor','texturemap');
Then you can manipulate h.XData and h.YData to animate changing the mapping.
for t = 0:.05:1
[newx, newy] = mymapping(x,y,t);
h.XData = newx;
h.YData = newy;
drawnow
end

Answer 2

With the following code I can generate a sequence of images that could be considered as a transformation or animation using the linear equation (1-t) * 1 + t * (1./(a.*(x+1i.*y))); But it is in no way that what is seen in CH56_Clip4.mov. Additionally t = 0.00 delivers a monochrome result.
[h_i, w_i, ~] = size(img);
% INITIALIZE IN-MESHGRID & NORMALIZE [0 1]
[xi,yi] = meshgrid(linspace(0,w_i+1,w_i),linspace(0,h_i+1,h_i));
xi = (xi/w_i); yi = (yi/h_i);
% INITIALIZE OUT-MESHGRID & NORMALIZE [-1 1]
[xo,yo] = meshgrid(1:w_i, 1:h_i);
xo = (xo/w_i)*2 - 1; yo = (yo/h_i)*2 - 1;
% GENERATE SURFACE
h = surf(xo, yo, zeros(w_i, h_i), img, 'FaceColor', 'texturemap');
% INITIALIZE OUT-IMAGE
img_out_t = zeros(h_i, w_i, 3);
for t = 0.00:0.1:1.0
% DISTORTION, SEPARATE & NORMALIZE [0 1]
fxiy = (1-t) * 1 + t * (1./(1.2.*(xo+1i.*yo)));
h.XData = real(fxiy); xo1 = h.XData;
h.YData = imag(fxiy); yo1 = h.YData;
xo1 = (xo1+1)/2; yo1 = (yo1+1)/2;
% INTERPOLATION
for k = 1:3; img_out_t(:,:,k) = uint8(interp2(xi, yi, img(:,:,k), xo1, yo1, interpol_meth)); end; imwrite(uint8(img_out_t), ['img_out_t_', num2str(t), '.jpg'], 'jpg', 'Quality', 98);
end