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 In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...

from one

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

to another that preserves the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

s of figures.

Properties

If ''M'' and ''N'' are two Riemannian (or

pseudo-Riemannian In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which t ...

) surfaces, then an equiareal map ''f'' from ''M'' to ''N'' 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 In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...

''U'' on ''M''. * The

pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...

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 vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...

s ''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 A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by convert ...

of vectors and ''df'' denotes the

pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...

along ''f''.

Example

An example of an equiareal map, due to

Archimedes of Syracuse Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

, 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 is equiareal, but the converse is not true. In fact,

shear mapping In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin. This type of mappi ...

and

squeeze mapping In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping. For a fixed positive real number , th ...

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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

. 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 Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

In map projections

In the context of geographic maps, a

map projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...

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