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Equiareal map

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In differential geometry, an equiareal map, sometimes called an authalic map, is a smooth map from one surface to another that preserves the areas of figures.

Properties

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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:

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

Example

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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.

Linear transformations

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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, 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 multiplies areas by the absolute value of its determinant |adbc|.

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 map projections

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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 and . For examples of such projections, see equal-area map projection.

See also

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References

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