Möbius Transformations helps visualize angle-preserving self-maps of the unit sphere in \(3\)-space. In Spring 2020, complex analysis student Amber Wu asked about visualizing conformal self-maps of the sphere. Two years later, while supervising a student summer project, I had the time to write this. The image below is a screenshot. Please visit the link for the live software.
About the Math (tl; dr): If we pick three distinct points on the unit sphere, there exists a unique conformal self-map of the sphere sending the zeroth point to the south pole, the first point to latitude \(0\) and longitude \(0\), and the second point to the north pole. The program visualizes these mappings by plotting the image of a geographic grid.
Instructions: Click in a form field. Either type in an extended complex number (such as 3-2i, or Inf), or use the q-a keys to increment/decrement the latitude of the selected point, or use the w-s keys to increment/decrement the longitude of the selected point. Holding down any of these four keys will move the plot through an abstract circle in the space of mappings. Note: Because we are manipulating the preimage rather than the image, the visual effect may not be as first expected.
Viewpoint: The sphere can be rotated in space using the arrow keys. Up and Down control pitch, while Left and Right control yaw.
About the Math in more detail: Let \((z, w)\) denote (complex) rectangular coordinates in \(\mathbf{C}^{2}\). Each non-vertical (complex) line through the origin has equation \(w = \alpha z\) for some complex slope \(\alpha\). In the sense of limits, the vertical line has infinite slope.
Further, the set of complex numbers together with one point at infinity, denoted \(\infty\), may be identified with the unit sphere in (real) \(3\)-space by stereographic projection. If \(z_{0}\), \(z_{1}\), and \(z_{\infty}\) are distinct slopes, it turns out there exists a unique angle-preserving self-map \(T\) of the sphere satisfying \(T(z_{0}) = 0\), \(T(z_{1}) = 1\), and \(T(z_{\infty}) = \infty\), given by the fractional linear formula \[ T(z) = \frac{(z – z_{0})(z_{1} – z_{\infty})}{(z – z_{\infty})(z_{1} – z_{0})}. \] By drawing a geographic grid (faint blue) and its image under \(T\) (bright green), we can visualize the effect of an angle-preserving self-map of the sphere.
For yet more detail, see the blog post Stereographic Projection and Möbius Transformations and the section on the projective line in the blog post Projective Geometry.