02 Oct 2021

*Tags: cartography
*

My paper “A variation on the Chamberlin trimetric map projection” was just published in Cartography and Geographic Information Science. If you don’t have institutional access, the preprint is on my github repo.

**The casual summary:** A compromise map projection is one that isn’t area-preserving or conformal (equal-angle). Conformal map projections have a lot of area distortion, and equal-area map projections have a lot of angle distortion. Compromise projections can have less noticable distortion than conformal or equal-area map projections. The concept of the just-noticable difference from psychophysics is relevant: for visual stimuli, the just-noticable difference is in the ballpark of 5%.

The Chamberlin trimetric map projection is a compromise map projection, useful for maps of continents and large regions. Without going into detail, it uses triangulation. It has a nice geometric construction, but the corresponding algebraic formula is long and fussy. It also lacks an easy-to-calculate inverse.

By changing how the triangulation is done very slightly, a projection with a much nicer algebraic formula was found. This was called the Matrix trimetric projection, since the formula involves the product of a 3x3 matrix with a vector. It takes about half the time to calculate as the Chamberlin projection. The formula can be inverted easily using a simple one-parameter Newton’s method iteration. The Matrix trimetric causes slightly more area and angle distortion than the Chamberlin, but it’s still in the ballpark of that just-noticable difference.

**The personal summary:** This was my COVID project. A good distraction from doomscrolling and compulsively reloading COVID dashboards, I suppose. I was intimidated by submitting an academic paper as someone who’s not affiliated with an institution any more, but it turned out to be a non-issue.

- 2023-11-04: Spherical area coordinates, and a derived triangle center
- 2023-07-29: Homogeneous coordinates on the sphere with vectors
- 2022-08-30: Scaling the Schwarz triangle function
- 2022-08-27: Edges in the image of the Schwarz triangle function
- 2022-07-20: Area-preserving swirling of the disk
- 2022-03-03: Some dubious ways to calculate conformal polyhedral projections
- 2021-08-31: Snyder's equal-area projection
- 2021-06-12: Perspective projections with vectors
- 2021-05-29: A (sort of) Euclidean triangle on a non-Euclidean surface
- 2021-05-08: Multilateration on the sphere with vectors