Homography can be used to “change the perspective” of an image (set of vectors). I have used homography for the satellite footprint in the computer simulations. For elevation angles smaller than , you can conveniently map transmitters inside the elliptical footprint to a circle for which the radially symmetric antenna pattern function can be used. The following homography matrix transforms an ellipse of parameters and to a circle of radius so that the right-hand side focus point maps to the origin.

Here is a GNU Octave code:

%%Changes ellipses with parameters a > b perspective to a sphere of radius r centered in origo. Vectors to be transformed are given in 2x1000 matrix refc. function points = homography(refc, a,b,r) c = sqrt(a^2 - b^2); %%Construct the homography matrix. X1 = [[-a -0 -1 0 0 0 a*r 0*r r]; [0 0 0 -a -0 -1 a*0 0*0 0]]; X2 = [[-c -b^2/a -1 0 0 0 c*0 b^2/a*0 0]; [0 0 0 -c -b^2/a -1 c*r b^2/a*r r]]; X3 = [[a 0 -1 0 0 0 -a*-r 0*-r -r]; [0 0 0 a -0 -1 -a*0 0*0 0]]; X4 = [[-c b^2/a -1 0 0 0 c*0 -b^2/a*0 0]; [0 0 0 -c b^2/a -1 c*-r b^2/a*r -r]]; P = [X1; X2; X3; X4]; [U,S,V] = svd(P); %Singular value composition. h = V(:,9); H = reshape(h, 3, 3)'; points = []; for point = refc homopoint = [point; 1]; %Point presented in homogeneous coordinates. homopoint = H*homopoint; homopoint = [homopoint(1)/homopoint(3); homopoint(2)/homopoint(3)]; points = [points homopoint]; end figure(1) plot(points(1,:), points(2,:), 'b','linewidth',10); end

In the following, we are rotating a “pyramid”. It can be seen how the homography mapping can be interpreted as a change of perspective.

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