Three Ends (Parametrizations I)

Most computer algebra systems come with some capabilities to render parametrized surfaces in space. You usually specify three functions of two variables x and y and a rectangle in the (x,y)-plane, and are rewarded with an image.

 

This has limitations: The most complicated topology you can achieve this way is a torus. Things get tricky when you want to draw something that has more than two ends.

Besides being able to draw these surfaces at all, one would also like to use a conformal parametrization so that the images of the parameter lines become orthogonal in space. This helps us being illusioned, because, having grown up in environments full of right angles, we assume that any intersection happens at a right angle.

firstquadrant

This can be accomplished for 3-ended surfaces by moving the ends to -1, 1 and infinity (using a Möbius transformation), dividing the plane into quadrants, and mapping a rectangle to the first quadrant so that we get polar coordinates at 1 and infinity as shown above. This is done using

f(z) = \sqrt{e^z+1}

on a rectangle of the form [a,b] x [0,π]: The exponential function maps the rectangle to a half-annulus in the upper half plane centered at 0. We then shift the “hole” at 0 to 1 and take a square root which bends the 180º angle at 0 to a right angle. The only thing to remember is that we want to have a parameter line hitting the origin, because otherwise our parameter mesh will have a gap there.

This is one of the simpler explicit parametrizations and responsible for the images on this page.