Waldo Tobler’s hyperelliptical projections

#projections #d3.js #forebears

28 February 2018


Born in 1930, Waldo R. Tobler died on February 20, 2018. He was one of the great American figures of geography, best known for his “first law of geography”:

Everything is related to everything else, but near things are more related than distant things.”

During his career as a researcher and teacher at the University of Michigan, then at the University of California at Santa Barbara, he was one of the pioneers in computerization of cartography, and contributed to the mathematical study of geographical spaces, particularly through the analysis of factors of scale. Cellular, fractal, computational geography: all these approaches allow, through modelling or simulation, to evaluate certain hypotheses, to fill gaps in observations, or to formulate predictions (e. g. on the extent of forest fires in California). The idea was always to offer to the eye different images of the world, to stimulate reflection and nourish intuition.

This post isn’t meant as a complete panorama of Waldo R. Tobler’s work; we invite you to read instead the obituary article published by Keith C. Clarke in Cartography and Geographic Information Science.

For those who love vintage videos, here are two charming minutes made by Tobler in 1970, and recently exhumed from the archives: one of the first map animations in history.

Animated Cartography of Detroit
Waldo Tobler, ca. 1970.

Map projections

In the field of projections, two projects bear the stamp of Waldo Tobler:

Loximuthal projection

This projection gives the Earth an egg shape, the particularity of which is that the loxodromes (that is, the lines travelled by a traveller who follows a constant course on his compass) are represented by straight lines. We can, below, play with this projection by adjusting the reference parallel. The projection is available for D3 under the name d3.geoLoximuthal().

The loximuthal projection
Waldo R. Tobler, 1966. Implemented in D3 by Jason Davies.

Hyperelliptical projections

This family of projections, published in 1973, is based on a form called “super-ellipse” or "Lamé curve", which generalizes the circle formula ( + = 1) in (xk + yk = 1).

This formula is trendy in design because it creates squared shapes with rounded edges and continuous curvature (called squircle, a contraction of square and circle).

Tobler showed how to put a world map inside this curve, and calculated the deformations of this projection. As a modest tribute, we have implemented it in D3.

Hyperelliptical projection
parameters α=0, k=2.5, γ=1.183136
Waldo R. Tobler, 1973. Implemented in D3 by Philippe Rivière.

This family of projections will soon be available in d3-geo-projection under the name d3.geoHyperelliptical(), and can be modified with the options .k(), .alpha(), .gamma().

Finally, here are some maps made possible by this family of projections:

↬ Philippe Rivière.