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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the polar sine generalizes the
sine function 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 that is opp ...
of
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
to the
vertex angle In geometry, a vertex is an angle (shape) associated with a vertex of an n-dimensional polytope. In two dimensions it refers to the angle formed by two intersecting lines, such as at a "corner" (vertex) of a polygon. In higher dimensions there ca ...
of a
polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
. It is denoted by psin.


Definition


''n'' vectors in ''n''-dimensional space

Let v1, ..., v''n'' (''n'' ≥ 2) be non-zero
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
s in ''n''-dimensional space (R''n'') that are directed from a
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...
of a parallelotope, forming the edges of the parallelotope. The polar sine of the vertex angle is: : \operatorname(\mathbf_1,\dots,\mathbf_n) = \frac, where the numerator is the
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 ...
:\begin \Omega & = \det\begin\mathbf_1 & \mathbf_2 & \cdots & \mathbf_n \end = \begin v_ & v_ & \cdots & v_ \\ v_ & v_ & \cdots & v_ \\ \vdots & \vdots & \ddots & \vdots \\ v_ & v_ & \cdots & v_ \\ \end \end equal to the hypervolume of the parallelotope with vector edges : \begin \mathbf_1 &= (v_, v_, \dots, v_)^T \\ \mathbf_2 &= (v_, v_, \dots, v_)^T \\ & \,\,\,\vdots \\ \mathbf_n &= (v_, v_, \dots, v_)^T, \\ \end and in the denominator the ''n''-fold
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
:\Pi = \prod_^n \, \mathbf_i\, of the
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
s , , v''i'', , of the vectors equals the hypervolume of the ''n''-dimensional
hyperrectangle In geometry, an orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions. A necessary and sufficient condition is that it is congruent to the Cartesian product of intervals. If all of ...
, with edges equal to the magnitudes of the vectors , , v1, , , , , v2, , , ... , , v''n'', , (not the vectors themselves). Also see Ericksson. The parallelotope is like a "squashed hyperrectangle", so it has less hypervolume than the hyperrectangle, meaning (see image for the 3d case): :\Omega \leq \Pi \implies \frac \leq 1 and since this ratio can be negative, psin is always bounded between −1 and +1 by the
inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...
: :-1 \leq \operatorname(\mathbf_1,\dots,\mathbf_n) \leq 1,\, as for the ordinary sine, with either bound only being reached in case all vectors are mutually
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
. In the case ''n'' = 2, the polar sine is the ordinary
sine 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 that is oppo ...
of the angle between the two vectors.


In higher dimensions

A non-negative version of the polar sine which works in any -dimensional space () can be defined using the Gram determinant. The numerator is given as : \Omega = \sqrt \,, where the superscript T indicates
matrix transposition In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The t ...
. In the case ''m'' = ''n'' this is equivalent to 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 the definition given previously.


Properties


Interchange of vectors

The polar sine changes sign whenever two vectors are interchanged, due to the antisymmetry of row-exchanging in the determinant; however, its absolute value will remain unchanged. :\begin \Omega & = \det\begin\mathbf_1 & \mathbf_2 & \cdots & \mathbf_i & \cdots & \mathbf_j & \cdots & \mathbf_n \end \\ & = -\!\det\begin\mathbf_1 & \mathbf_2 & \cdots & \mathbf_j & \cdots & \mathbf_i & \cdots & \mathbf_n \end \\ & = -\Omega \end


Invariance under scalar multiplication of vectors

The polar sine does not change if all of the vectors v1, ..., v''n'' are scalar-multiplied by positive constants ''ci'', due to
factorization In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...
: \begin \operatorname(c_1 \mathbf_1,\dots, c_n \mathbf_n) & = \frac \\ pt& = \frac \cdot \frac \\ pt& = \operatorname(\mathbf_1,\dots, \mathbf_n). \end If an
odd number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...
of these constants are instead negative, then the sign of the polar sine will change; however, its absolute value will remain unchanged.


Vanishes with linear dependencies

If the vectors are not
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
, the polar sine will be zero. This will always be so in the
degenerate case In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. ...
that the number of dimensions is strictly less than the number of vectors .


Relationship to pairwise cosines

The cosine of the angle between two non-zero vectors is given by :\cos(\mathbf_1, \mathbf_2) = \frac\, using the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
. Comparison of this expression to the definition of the absolute value of the polar sine as given above gives: :\left, \operatorname(\mathbf_1, \ldots, \mathbf_n)\^2 = \det\!\left[\begin 1 & \cos(\mathbf_1, \mathbf_2) & \cdots & \cos(\mathbf_1, \mathbf_n) \\ \cos(\mathbf_2, \mathbf_1) & 1 & \cdots & \cos(\mathbf_2, \mathbf_n) \\ \vdots & \vdots & \ddots & \vdots \\ \cos(\mathbf_n, \mathbf_1) & \cos(\mathbf_n, \mathbf_2) & \cdots & 1 \\ \end\right]. In particular, for , this is equivalent to :\sin^2(\mathbf_1, \mathbf_2) = 1 - \cos^2(\mathbf_1, \mathbf_2)\,, which is the Pythagorean theorem.


History

Polar sines were investigated by
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in the 18th century.


See also

*
Trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
*
List of trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
*
Solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...
*
Simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
*
Law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and a ...
*
Cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
and
Seven-dimensional cross product In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in a vector also in . Like the cross product in three dimensions, the seven-dim ...
*
Graded algebra In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
*
Exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
*
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 ...
*
Volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ap ...
*
Measure (mathematics) In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
*
Product integral A product integral is any product-based counterpart of the usual sum-based integral of calculus. The first product integral ('' Type I'' below) was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equatio ...


References


External links

* {{MathWorld, PolarSine, Polar Sine Polytopes Trigonometry