In
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'',
where 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
:
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
:
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
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 R
2, in the obvious way in R
3, the requirement above then is weakened to:
:
for some not depending on
and
.
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 ...
.
See also
*
Jacobian matrix and determinant
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