In differential geometry, an equiareal map is a smooth map from one surface to another that preserves the area of figures. If M and N are two surfaces in the Euclidean space R3, 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 μN on N is equal to μM, the area element on M.
- At each point p of M, and tangent vectors v and w to M at p,
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.
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 R2, in the obvious way in R3, 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.