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In mathematics, the signed area or oriented area of a region of an affine plane is its
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
with orientation specified by the positive or negative sign, that is "plus" () or "minus" (). More generally, the signed area of an arbitrary surface region is its
surface area The surface area (symbol ''A'') 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 d ...
with specified orientation. When the boundary of the region is a simple curve, the signed area also indicates the orientation of the boundary.


Planar area


Polygons

The mathematics of ancient Mesopotamia,
Egypt Egypt ( , ), officially the Arab Republic of Egypt, is a country spanning the Northeast Africa, northeast corner of Africa and Western Asia, southwest corner of Asia via the Sinai Peninsula. It is bordered by the Mediterranean Sea to northe ...
, and
Greece Greece, officially the Hellenic Republic, is a country in Southeast Europe. Located on the southern tip of the Balkan peninsula, it shares land borders with Albania to the northwest, North Macedonia and Bulgaria to the north, and Turkey to th ...
had no explicit concept of
negative number In mathematics, a negative number is the opposite (mathematics), opposite of a positive real number. Equivalently, a negative number is a real number that is inequality (mathematics), less than 0, zero. Negative numbers are often used to represe ...
s or signed areas, but had notions of
shape A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...
s contained by some boundary lines or curves, whose areas could be computed or compared by pasting shapes together or cutting portions away, amounting to addition or subtraction of areas. This was formalized in Book I of Euclid's ''Elements'', which leads with several common notions including "if equals are added to equals, then the wholes are equal" and "if equals are subtracted from equals, then the remainders are equal" (among planar shapes, those of the same area were called "equal"). The propositions in Book I concern the properties of
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s and
parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...
s, including for example that parallelograms with the same base and in the same parallels are equal and that any triangle with the same base and in the same parallels has half the area of these parallelograms, and a
construction Construction are processes involved in delivering buildings, infrastructure, industrial facilities, and associated activities through to the end of their life. It typically starts with planning, financing, and design that continues until the a ...
for a parallelogram of the same area as any "rectilinear figure" (
simple polygon In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. The ...
) by splitting it into triangles. Greek geometers often compared planar areas by quadrature (constructing a
square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
of the same area as the shape), and Book II of the ''Elements'' shows how to construct a square of the same area as any given polygon. Just as negative numbers simplify the solution of algebraic equations by eliminating the need to flip signs in separately considered cases when a quantity might be negative, a concept of signed area analogously simplifies geometric computations and proofs. Instead of subtracting one area from another, two signed areas of opposite orientation can be added together, and the resulting area can be meaningfully interpreted regardless of its sign. For example, propositions II.12–13 of the ''Elements'' contain a geometric precursor of the
law of cosines In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see ...
which is split into separate cases depending on whether the angle of a triangle under consideration is obtuse or acute, because a particular rectangle should either be added or subtracted, respectively (the
cosine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
of the angle is either negative or positive). If the rectangle is allowed to have signed area, both cases can be collapsed into one, with a single proof (additionally covering the right-angled case where the rectangle vanishes). As with the unoriented area of simple polygons in the ''Elements'', the oriented area of
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
s in the affine plane (including those with holes or self-intersections) can be conveniently reduced to sums of oriented areas of triangles, each of which in turn is half of the oriented area of a parallelogram. The oriented area of any polygon can be written as a signed
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
coefficient (the ''signed area'' of the shape) times the oriented area of a designated polygon declared to have unit area; in the case of the
Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
, this is typically a
unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinat ...
. Among the computationally simplest ways to break an arbitrary polygon (described by an ordered list of vertices) into triangles is to pick an arbitrary origin point, and then form the oriented triangle between the origin and each pair of adjacent vertices in the triangle. When the plane is given a
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
, this method is the 18th century shoelace formula.


Curved shapes

The ancient Greeks had no general method for computing areas of shapes with curved boundaries, and the quadrature of the circle using only finitely many steps was an unsolved problem (proved impossible in the 19th century). However,
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
exactly computed the
quadrature of the parabola ''Quadrature of the Parabola'' () is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions regarding parabolas, culminating in two proofs showing t ...
via the
method of exhaustion The method of exhaustion () is a method of finding the area of a shape by inscribing inside it a sequence of polygons (one at a time) whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the differ ...
, summing infinitely many triangular areas in a precursor of modern integral calculus, and he approximated the quadrature of the circle by taking the first few steps of a similar process.


Integrals

The
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
of a
real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
can be imagined as the signed area between the x-axis and the curve y = f(x) over an interval . The area above the x-axis may be specified as positive (), and the area below the x-axis may be specified as negative (). The ''negative area'' arises in the study of
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
as signed area below the curve y = 1/x for , that is: \ln x = \int_1 ^x \frac = - \int_x^1 \frac < 0. In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, the sign of the area of a region of a surface is associated with the orientation of the surface. The area of a set in differential geometry is obtained as an integration of a
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
: \mu(A) = \int_A dx \wedge dy, where and are differential 1-forms that make the density. Since the wedge product has the anticommutative property, . The density is associated with a planar orientation, something existing locally in a manifold but not necessarily globally. In the case of the natural logarithm, obtained by integrating the area under the hyperbola , the density is positive for , but since the integral \int_1^x \frac is anchored to , the orientation of the -axis is reversed in the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...
. For this integration, the () orientation yields the opposite density to the one used for . With this opposite density the area, under the hyperbola and above the unit interval, is taken as a negative area, and the natural logarithm consequently is negative in this domain.


Determinants

Signed areas were associated with
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
s by
Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
in 1908.
Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
, translators E.R. Hendrick & C.A. Noble (1939) 908''Elementary Mathematics from an Advanced Standpoint – Geometry'', third edition, pages 3, 10, 173,4
When a triangle is specified by three points, its area is: \frac \beginx_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\x_3 & y_3 & 1 \end. For instance, when (x_3,\ y_3) = (0,0) , then the area is given by \frac. To consider a sector area bounded by a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
, it is approximated by thin triangles with one side equipollent to which have an area \frac \begin 0 & 0 & 1 \\ x & y & 1 \\ x+dx & y+dy & 1 \end = \frac (x\ dy - y\ dx). Then the "area of the sector between the curve and two radius vectors" is given by \frac \int_1 ^2 (x\ dy - y\ dx). For example, the ''reverse'' orientation of the unit hyperbola is given by x = \cosh t, ~ y = -\sinh t . Then x \ dy - y \ dx = - \cosh^2 t \ dt + \sinh^2 t \ dt = - dt , so the area of the
hyperbolic sector A hyperbolic sector is a region (mathematics), region of the Cartesian plane bounded by a hyperbola and two ray (geometry), rays from the origin to it. For example, the two points and on the Hyperbola#Rectangular hyperbola, rectangular hyperbol ...
between and is \frac \int_0^\theta dt = \left. \frac \_0^\theta = -\frac, giving a negative hyperbolic angle as a negative sector area.


Postnikov equivalence

Mikhail Postnikov's 1979 textbook ''Lectures in Geometry'' appeals to certain geometric transformations – described as functions of coordinate pairs (x, y) – to express "freely floating area elements". A ''
shear mapping In plane geometry, a shear mapping is an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed distance function, signed distance from a given straight line, line parallel (geometry), paral ...
'' is either of: \begin (x,y) &\to (x,\ y + kx), \\ (x,y) &\to (x + ky,\ y) \end for any real number , while a ''
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 (mathematics), rotation or shear mapping. For a fixed p ...
'' is (x,y) \to (\lambda x,\ y/\lambda) for any positive real number . An area element is related () to another if one of the transformations results in the second when applied to the first. As an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
, the area elements are segmented into
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of related elements, which are Postnikov bivectors. Proposition: If (a_1, b_1) = (ka + \ell b, \ k_1 a + \ \ell_1 b) and : \delta = k \ell_1 - \ell k_1 \ne 0 , then (a, b) \thicksim (a_1, \ (\frac) b_1) . : proof: (a,\ b) \thicksim (a + \frac b, \ b) shear mapping : \thicksim (ka + \ell b, \ \tfrac b) squeeze mapping : \thicksim (ka + \ell b, \ \tfrac b + \frac (ka + \ell b) ) shear mapping : = (ka + \ell b, \ \frac (ka + (\frac + \ell) b) : = (ka + \ell b, \ \frac (k_1a + \ell_1 b) ) \ \ \text\ \ \ell + \frac = \frac \ell_1 : = (a_1, \ \frac b_1) .


See also

*
Vector area In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an oriented area in three dimensions. Every bounded surface in three dimensions can be associated with a ...
*
Volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of t ...
* Signed distance *
Signed measure In mathematics, a signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to acquire sign. Definition There are two slightly different concepts of a signed measure, de ...
* Signed volume


References


External links

* {{cite web , last1=Kleitman , first1=Daniel , authorlink=Daniel Kleitman, title=Chapter 15: Areas and Volumes of Parallel Sided Figures; Determinants, url=https://math.mit.edu/~djk/calculus_beginners/chapter15/contents.html , website=Kleitman's Homepage , publisher=MIT Math Department Homepage Planes (geometry) Measure theory Area Sign (mathematics)