Equiareal map

Indifferential geometry,anequiareal map,sometimes called anauthalic map,is asmooth mapfrom onesurfaceto another that preserves theareasof figures.

Properties

IfMandNare twoRiemannian(orpseudo-Riemannian) surfaces, then an equiareal mapffromMtoNcan be characterized by any of the following equivalent conditions:

• Thesurface areaoff(U) is equal to the area ofUfor everyopen setUonM.
• Thepullbackof thearea elementμ_NonNis equal toμ_M,the area element onM.
• At each pointpofM,andtangent vectorsvandwtoMatp,

$${\bigl |}df_{p}(v)\wedge df_{p}(w){\bigr |}=|v\wedge w|\,$$

where${\textstyle \wedge }$denotes the Euclideanwedge productof vectors anddfdenotes thepushforwardalongf.

Example

An example of an equiareal map, due toArchimedes of Syracuse,is the projection from the unit spherex²+y²+z²= 1to the unit cylinderx²+y²= 1outward from their common axis. An explicit formula is

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

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

Linear transformations

EveryEuclidean isometryof theEuclidean planeis equiareal, but theconverseis not true. In fact,shear mappingandsqueeze mappingare counterexamples to the converse.

Shear mapping takes a rectangle to a parallelogram of the same area. Written in matrix form, a shear mapping along thex-axis is

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

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

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

A linear transformation${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$multiplies areas by theabsolute valueof itsdeterminant|ad–bc|.

Gaussian eliminationshows that every equiareal linear transformation (rotationsincluded) can be obtained by composing at most two shears along the axes, a squeeze and (if the determinant is negative), areflection.

In map projections

Main article:Equal-area map projection

In the context ofgeographic maps,amap projectionis calledequal-area,equivalent,authalic,equiareal,orarea-preserving,if areas are preserved up to a constant factor; embedding the target map, usually considered a subset ofR²,in the obvious way inR³,the requirement above then is weakened to:

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

for someκ> 0not depending on$v$and$w$. For examples of such projections, seeequal-area map projection.

See also

• Jacobian matrix and determinant

References

• Pressley, Andrew (2001),Elementary differential geometry,Springer Undergraduate Mathematics Series, London: Springer-Verlag,ISBN978-1-85233-152-8,MR1800436