Map Projections

Deducing the Kavrayskiy VII Projection

Consider Kavrayskiy (Kavraisky)'s VII, a map projection I will describe, in the equatorial aspect, as: pseudocylindrical, poles half as long as the Equator, equidistant parallels, meridian 120° contained in a circle centered on the map. What does that mean?

pseudocylindrical
in the equatorial aspect only, all parallels are horizontal straight lines, while meridians are arbitrary curves
poles half as long as the equator
from the definition, the poles in pseudocylindrical projections are either points or straight lines ("polelines"). Compared with "pointed-polar" projections, "flat-polar" designs usually have lesser shape distortion in high latitudes. Most flat-polar maps have simple pole length/Equator length ratios, like or
equidistant parallels
all but one equal-area pseudocylindrical projections have variable distance between parallels; in constrast, the non-equal-area Kavrayskiy VII has constant parallel spacing, which does not of course mean that the scale along its meridians is identical or even constant. Without further constraints, making the central meridian alone a standard line
meridian 120° contained in a circle centered on the map
in every pseudocylindrical projection the scale along any parallel is constant. Therefore, all meridians have similar shapes, except the central one (considered 0° here) which is always straight. Shapes are affected by the longitude. In this case, if the meridian at 120° (i.e., ) is an arc of circle, all others but the central one are elliptical arcs, flattened towards the central meridian, elongated towards the map's boundary

Top right quadrant of Kavrayskiy VII map

Let us sketch the northeastern quadrant of the Kavrayskiy VII projection. Let be half the poleline's length, and its distance from the Equator; due to the equidistant property. For a given latitude , let be the abscissa of the circular 120° meridian, and the abscissa of the 180° meridian, the fundamental boundary meridian.

Because true distances are linear along each pseudocylindrical parallel, both and the general abscissa at any given latitude are proportional to :

The radius of the reference circle follows immediately. And from the Pithagorean theorem,

Half the length of the Equator is:

Again from Pithagoras's theorem,

Kavrayskiy VII map

Easy to understand and compute, Kavrayskiy's 7th projection has no special properties, preserving neither shapes nor areas. It simply has a good and balanced appearance, which for many purposes is often good enough.

www.progonos.com/furuti/MapProj/Normal/CartHow/HowKav7/howKav7.html — August  5, 2018
Copyright © 2012 Carlos A. Furuti