differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

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

Properties

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: * The

surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...

of ''f''(''U'') is equal to the area of ''U'' for every open set ''U'' on ''M''. * The pullback of the

area element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV = ...

''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'',

\bigl, df_p(v)\wedge df_p(w)\bigr,  = , v\wedge w, \,

where $\wedge$ denotes the Euclidean wedge 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 to the unit cylinder outward from their common axis. An explicit formula is :

f(x,y,z) = \left(\frac, \frac, z\right)

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

Linear transformations

Every

Euclidean isometry In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformation ...

of the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

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 -axis is :

\begin1 & v \\ 0 & 1 \end \,\beginx\\y \end =   \beginx+vy\\y \end.

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\lambda & 0 \\ 0 & 1/\lambda  \end\,\beginx\\y \end = \begin\lambda x\\ y/\lambda.\end

A linear transformation

\begina & b \\ c & d \end

multiplies areas by the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

of its determinant .

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

shows that every equiareal linear transformation (

rotations Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

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

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: :

, df_p(v)\times df_p(w), =\kappa, v\times w,

for some not depending on

v

and

w

. For examples of such projections, see

equal-area map projection In cartography, an equal-area projection is a map projection that preserves area measure, generally distorting shapes in order to do that. Equal-area maps are also called equivalent or authalic. An equal-area map projection cannot be conformal, no ...

References

*{{Citation , last1=Pressley , first1=Andrew , title=Elementary differential geometry , publisher=Springer-Verlag , location=London , series=Springer Undergraduate Mathematics Series , isbn=978-1-85233-152-8 , mr=1800436 , year=2001 Differential geometry Functions and mappings

Properties

Example

Linear transformations

In map projections

See also

References