A variation on the Chamberlin trimetric map projection

ABSTRACT

We present a variation on the Chamberlin trimetric map projection. This new projection, which we call the matrix trimetric projection, consists of a linear transformation of the squares of the distances between a given point and three control points. The formula of the forward projection is simpler than the Chamberlin projection, and admits an inverse formula which requires numerical iteration of only one parameter. We make comparisons between the two projections using a representative list of control points. The Chamberlin trimetric projection outperforms the matrix trimetric projection on measures of angle deformation and area deformation, but the opposite is true for a measure of distance deformation, and the difference between the results of the projections is small over all measures. The forward Matrix trimetric projection can be calculated in half the time of the Chamberlin trimetric projection. We conclude that the matrix trimetric projection is a viable alternative to the Chamberlin trimetric projection, especially if an inverse is required or speed is important.

KEYWORDS:

• Chamberlin trimetric
• matrix trimetric
• projection
• cartography
• distortion
• distance
• triangulation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

A software implementation of the MTP, as well as code used to produce the calculations and figures in this text, is available on the author’s Github site https://github.com/brsr/trimetric. (Recht, 2021)

Figures in this text were created using the Python packages matplotlib (Hunter, 2007) and GeoPandas (Jordahl et al., 2020), and their dependencies including PROJ (PROJ contributors, 2019) and PyProj (Snow et al., 2020) as referenced earlier. Land mass shape data come from Patterson and Kelso (2020).