# Equiareal map

In differential geometry, an **equiareal map** is a smooth map from one surface to another that preserves the areas of figures. If *M* and *N* are two surfaces in the Euclidean space **R**^{3}, then an equi-areal map *f* can be characterized by any of the following equivalent conditions:

- The surface area of
*f*(*U*) is equal to the area of*U*for every open set*U*on*M*. - The pullback of the area element
*μ*_{N}on*N*is equal to*μ*_{M}, the area element on*M*. - At each point
*p*of*M*, and tangent vectors*v*and*w*to*M*at*p*,

- where × denotes the Euclidean cross 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 *x*^{2} + *y*^{2} + *z*^{2} = 1 to the unit cylinder *x*^{2} + *y*^{2} = 1 outward from their common axis. An explicit formula is

for (*x*, *y*, *z*) a point on the unit sphere.

In the context of geographic maps, a map projection is called **equiareal**, **area-preserving**, or more commonly **equi-area**, if areas are preserved up to a constant factor; embedding the target map, usually considered a subset of **R**^{2}, in the obvious way in **R**^{3}, the requirement above then is weakened to:

for some *κ* > 0 not depending on and .
For examples of such projections, see Equal-area map projections. Linear equi-areal maps are 2 × 2 real matrices making up the group SL(2, **R**) of special linear transformations.