This post will present the conformal map between the interior of an ellipse and the unit disk.

Given an ellipse centered at the origin with semi-major axis *a* and semi-minor axis *b*. Will will assume without loss of generality that *a*² – *b*² = 1 and so the foci are at ±1.

Hermann Schwarz published the conformal map from the ellipse to the unit disk in 1869 [1, 2].

The map is given by

where sn is the Jacobi elliptic function with parameter *k*². The constants *k* and *K* are given by

where θ_{2} and θ_{3} are theta constants, the value so the theta functions θ_{2}(*z*, *q*) and θ_{3}(*z*, *q*) at *z* = 1.

Conformal maps to the unit disk are unique up to rotation. The map above is the unique conformal map preserving orientation:

## Inverse map

The inverse of this map is given by

The inverse of the sn function with parameter *m* can be written in terms of elliptic integrals.

where *F* is the incomplete elliptic integral of the first kind and *m* is the parameter of sn and the parameter of *F*.

## Plot

I wanted to illustrate the conformal map using an ellipse with aspect ratio 1/2. To satisfy *a*² – *b*² = 1, I set *a* = 2/√3 and *b* = 1/√3. The plot at the top of the post was made using Mathematica.

## Related posts

- NASA and conformal maps
- Comparing Jacobi functions and trig functions
- Conformal mapping and Laplace’s equation
- Numerically evaluate a theta function

[1] H. A. Schwarz, Über eigige Abbildungsaufgaben, Journal für di reine und angew. Matheamatik, vol 70 (1869), pp 105–120

[2] Gabor Szegö. Conformal Mapping of the Interior of an Ellipse onto a Circle. The American Mathematical Monthly, 1950, Vol. 57, No. 7, pp. 474–478