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The homotopy between a Roman surface and a Boy surface can be given by the following equations [1]:

for u [-PI/2, PI/2] and v [0, PI] and alpha varies from 0 (Roman surface) through 1 (Boy surface).

The following figures show the transformation between the two:

And here is the code for generating this (you can also use this to render an animation).

clear;
stacksize (10000000);
num_points = 50;
u = linspace (-%pi/2, %pi/2, num_points);
v= linspace (0, %pi, num_points);
[U, V] = meshgrid (u, v);
//alpha = 0: Roman surface; alpha = 1: Boy surface;
alpha = linspace(0, 1, 25);
for i = 1:length(alpha)
x = ((sqrt(2).*cos(2*U).*cos(V).*cos(V))+cos(U).*sin(2*V))./(2-alpha(i).*sqrt(2).*sin(3*U).*sin(2*V));
y = ((sqrt(2).*sin(2*U).*cos(V).*cos(V))-sin(U).*sin(2*V))./(2-alpha(i).*sqrt(2).*sin(3*U).*sin(2*V));
z = (3*cos(V).*cos(V))./(2-alpha(i)*sqrt(2).*sin(3*U)*sin(2*V));
surf(x, y, z,’edgecol’,'gre’);
end

[1] Courtesy: Roman surface (Mathworld)

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