# NASA and conformal maps

A couple years ago I wrote about how NASA was interested in regions bounded by curves of the form

For example, here’s a plot for A = 2, B = 1, α = 2.5 and β = 6.

That post mentioned a technical report from NASA that explains why these shapes are important in application, say for beams and bulkheads, and gives an algorithm for finding conformal maps of the unit disk to these shapes. These shapes are also used in graphic design, such as squircle buttons on iPhones.

Note that these shapes are not rounded rectangles in the sense of a rectangles modified only at the corners. No segment of the sides is perfectly flat. The curvature smoothly decreases as you move away from the corners rather than abruptly jumping to 0 as it does in a rectangle rounded only at the corners. Maybe you could call these shapes continuously rounded rectangles.

## Conformal maps

Conformal maps are important because they can map simple regions to more complicated regions in way that locally preserves angles. NASA might want to solve a partial differential equation on a shape such as the one above, say to calculate the stress in a cross section of a beam, and use conformal mapping to transfer the equation to a disk where the calculations are easier.

## Coefficients

The NASA report includes a Fortran program that computes the coefficients for a power series representation of the conformal map. All the even coefficients are zero by reasons of symmetry.

Coefficients are reported to four decimal places. The closer the image is to a circle, i.e. the closer α and β are to 2 and the closer A is to B, the fewer non-zero coefficients the transformation has. You could think of the number of non-zero coefficients as a measure how hard the transformation has to work to transform the disk into the desired region.

There are interesting patterns in the coefficients that I would not have noticed if I had not had to type the coefficients into a script for the example below. Maybe some day I’ll look more into that.

## Conformal map example

The following example uses A = 2, B = 1, and α = β = 5.