Equiareal map (mathematics)

In differential geometry, an equiareal map (or equi-areal map) is a smooth map from one surface to another that preserves the areas of figures.

Properties

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 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,

${\displaystyle |df_{p}(v)\times df_{p}(w)|=|v\times w|\,}$

where × denotes the Euclidean cross product of vectors and df denotes the pushforward along f.

Example

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

${\displaystyle f(x,y,z)=\left({\frac {x}{\sqrt {x^{2}+y^{2}}}},{\frac {y}{\sqrt {x^{2}+y^{2}}}},z\right)}$

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

Linear transformations

Every Euclidean isometry of the Euclidean plane is equiareal, but the converse is not true. In fact, shear mapping and squeeze mapping are counterexamples to the converse.

Shear mapping takes a rectangle to a parallelogram of the same area. Written in matrix form, the mapping is

${\displaystyle (x,y){\begin{pmatrix}1&0\\v&1\end{pmatrix}}\ =\ (x+vy,\ y).}$

A common application is in classical kinematics where y is a temporal value (time). In this context the shear is a Galilean transformation.

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

${\displaystyle (x,y){\begin{pmatrix}\lambda &0\\0&1/\lambda \end{pmatrix}}\ =\ (\lambda x,y/\lambda ).}$

In relativistic kinematics the speed of light c is a supremum for velocity. Hyperbolas xy = k are stable under squeezing. Velocity, represented on a hyperbola and called rapidity, is transformed by "hyperbolic rotation" within (–c, c).

According to exterior algebra, a linear transformation ${\displaystyle {\begin{pmatrix}a&c\\b&d\end{pmatrix}}}$ multiplies area by the magnitude of its determinant ad – bc. That rotation, shear, and squeeze exhaust the types of equiareal linear transformations is shown at 2 × 2 real matrices as complex numbers. These mappings form the special linear group SL(2,R).

In map projections

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:

${\displaystyle |df_{p}(v)\times df_{p}(w)|=\kappa |v\times w|}$

for some κ > 0 not depending on ${\displaystyle v}$ and ${\displaystyle w}$. For examples of such projections, see equal-area map projection.

See also

• Jacobian matrix and determinant

References

• Pressley, Andrew (2001), Elementary differential geometry, Springer Undergraduate Mathematics Series, London: Springer-Verlag, ISBN 978-1-85233-152-8, MR 1800436