The Upper Half Disk (Parametrizations 2)

One of the text book examples of conformal maps is given by the complex function f(z) = z+1/z which maps the upper half disk bijectively to the lower half plane. This is the main ingredient for the following problem:

We would like to have a conformal map from a rectangle to a half disk (say of radius r) so that the left and right sides of the rectangle are mapped to half circular arcs around two points a and b.

The inverse of this map can be obtained by composing the text book map above with a Möbius transformation that sends the images of a and b to 0 and infinity, and then use polar coordinates on the upper half plane. Details are in this notebook.

What is this good for? Examples are 4-ended surfaces like the ones above that have sufficient symmetry that one can place the ends as four punctures on the real line, symmetric with respect to a circle.

This can also be useful more generally when existing symmetries cut the surface into pieces with four ends along the boundary of the pieces, as above.

Out of the Box

One of many ways to construct triply periodic minimal surfaces is via conjugate surfaces. You start with a polygon in space, solve the Plateau problem, construct the conjugate surface. Instead of straight edges, this conjugate surface piece will allow extension by reflecting across its boundary edges, and with a fair amount of luck, you obtain a nice triply periodic surface. There are two disadvantages of the Plateau method: You are numerically limited to minimizing Plateau solutions, and the whole approach gives little theoretical insight. Here is a variation of this approach:

We start with a minimal polygon inside a box with all edges perpendicular to the faces of the box. Reflections at the faces will produce 8 copies, which constitute a translational fundamental piece of a triply periodic surface. If we look at the boundary of the polygon in the vertical faces, we note that at the corners the Gauss map will be vertical. We encode this in a sequence of + and – signs. For the left boundary component in the example, we have two points with normal pointing (say) up, encoded by +. In the second component we first point down at the upper point, and then down at the lower point, encoded by +-. Both sequences give the symbol (++|+-).

The same information is also contained in the shaded rectangle above, with the red dots labeled a and b corresponding to the corners in the left boundary edge, and c and corresponding to those in the right edge. The entire rectangle then represents the torus quotient of the surface under the 180º rotation about the vertical axis. The vertices are the zeroes and poles of the Gauss map.

Together with the additional reflectional symmetries at the horizontal box faces (the vertical green lines in the rectangle), this information determines the Gauss map. The height differential on the quotient torus is just dz, so we have the entire Weierstrass representation of the surface, except that we do not know the values of the parameters a,b,c,d and τ.

A linear combination of the parameters a,b,c,d determines how the Gauss map rotates in horizontal symmetry planes. For 8-gons as above, one usually is then left with a 2-dimensional period problem, resulting in a 2-dimensional family of examples. This approach is useful for three reasons: One can use the Enneper-Weierstrass representation for theoretical and numerical purposes, investigate limits easily, and extend the method by forsaking the horizontal symmetries, as we will see at a later point.

This page shows examples for these seven types, you can find more under the individual surface pages, listed under the genus 5 box types section in the triply periodic minimal surfaces page.

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.