If M and N are two Riemannian (or pseudo-Riemannian) surfaces, then an equiareal map f from M to N can be characterized by any of the following equivalent conditions:
denotes the Euclidean wedge product of vectors and df denotes the pushforward along f. An example of an equiareal map, due to Archimedes of Syracuse, is the projection from the unit sphere x2 + y2 + z2 = 1 to the unit cylinder x2 + y2 = 1 outward from their common axis.
An explicit formula is for (x, y, z) a point on the unit sphere.
Shear mapping takes a rectangle to a parallelogram of the same area.
Written in matrix form, a shear mapping along the x-axis is Squeeze mapping lengthens and contracts the sides of a rectangle in a reciprocal manner so that the area is preserved.
Written in matrix form, with λ > 1 the squeeze reads A linear transformation
Gaussian elimination shows that every equiareal linear transformation (rotations included) can be obtained by composing at most two shears along the axes, a squeeze and (if the determinant is negative), a reflection.
In the context of geographic maps, a map projection is called equal-area, equivalent, authalic, equiareal, or area-preserving, if areas are preserved up to a constant factor; embedding the target map, usually considered a subset of R2, in the obvious way in R3, the requirement above then is weakened to: for some κ > 0 not depending on