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?

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.