Hyperbolic Geometry & Conformal Maps

Conformal Maps

Image of a hyperboloid.
Some notes about conformal maps. (2019)

Möbius Transformations

Some notes about Möbius transformations, which are especially nice conformal maps. Well, in 2D they're especially nice. In higher dimensions all conformal maps are Möbius transformations! (2019)

The Hyperboloid Model

Image of a hyperboloid.
The hyperboloid model is a useful model of hyperbolic space which naturally sits inside of (2+1) dimensional Minkowski space. (2019)

Hyperbolic Triangles

Image of a hyperboloid.
Notes about ideal hyperbolic triangles and how to represent ideal hyperbolic triangulations with the signpost datastructure. (2019)

Poincaré Disk

Image of a hyperboloid.
Interactive visualization of the Poincaré disk model of the hyperbolic plane. (2019)

Milnor's Lobachevsky Function

Milnor's Lobachevsky function is a strange, and strangely named, function which is surprisingly useful for discrete uniformization. (2019)

Penner Coordinates

Although ideal hyperbolic triangles have infinite perimeters, we can still assign them meaningful numbers that act like side lengths. These are called Penner coordinates. (2019)

Discrete Conformal Maps and Ptolemy Flips

Notes exploring the connections between discrete conformal maps, circumcircle-preserving projective maps, and Ptolemy flips, as described in Discrete Conformal Maps and Ideal Hyperbolic Polyhedra by Bobenko, Pinkall and Springborn and Ideal Polyhedra and Discrete Uniformization by Springborn. (2019)

Extremal Quasiconformal Maps

Image of a rectangle being stretched.
Some meandering thoughts about what it might mean to have a discrete quasiconformal map. (2023)