Projective frame

Last updated February 19, 2022

In mathematics, and more specifically in projective geometry, a projective frame or projective basis is a tuple of points in a projective space that can be used for defining homogeneous coordinates in this space. More precisely, in a projective space of dimension $n$ , a projective frame is a $n + 2$ -tuple of points such that no hyperplane contains $n + 1$ of them. A projective frame is sometimes called a simplex,^[1] although a simplex in a space of dimension $n$ has at most $n + 1$ vertices.

In this article, only projective spaces over a field $K$ are considered, although most results can be generalized to projective spaces over a division ring.

Let $P (V)$ be a projective space of dimension $n$ , where $V$ is a $K$ -vector space of dimension $n + 1$ . Let $p:V\setminus \{0\}\to \mathbf {P} (V)$ be the canonical projection that maps a nonzero vector $v$ to the corresponding point of $P (V)$ , which is the vector line that contains $v$ .

Every frame of $P (V)$ can be written as $\left(p(e_{0}),\ldots ,p(e_{n+1})\right),$ for some vectors $e_{0},\dots ,e_{n+1}$ of $V$ . The definition implies the existence of nonzero elements of $K$ such that $\lambda _{0}e_{0}+\cdots +\lambda _{n+1}e_{n+1}=0$ . Replacing $e_{i}$ by $\lambda _{i}e_{i}$ for $i\leq n$ and $e_{n+1}$ by $-\lambda _{n+1}e_{n+1}$ , one gets the following characterization of a frame:

n + 2

points of

P (V)

form a frame if and only if they are the image by

p

of a basis of

V

and the sum of its elements.

Moreover, two bases define the same frame in this way, if and only if the elements of the second one are the products of the elements of the first one by a fixed nonzero element of $K$ .

As homographies of $P (V)$ are induced by linear endomorphisms of $V$ , it follows that, given two frames, there is exactly one homography mapping the first one onto the second one. In particular, the only homography fixing the points of a frame is the identity map. This result is much more difficult in synthetic geometry (where projective spaces are defined through axioms). It is sometimes called the first fundamental theorem of projective geometry.^[2]

Every frame can be written as $(p(e_{0}),\ldots ,p(e_{n}),p(e_{0}+\cdots +e_{n})),$ where $(e_{0},\dots ,e_{n})$ is basis of $V$ . The projective coordinates or homogeneous coordinates of a point $p (v)$ over this frame are the coordinates of the vector $v$ on the basis $(e_{0},\dots ,e_{n}).$ If one changes the vectors representing the point $p (v)$ and the frame elements, the coordinates are multiplied by a fixed nonzero scalar.

Commonly, the projective space $P n (K) = P (K n +1)$ is considered. It has a canonical frame consisting of the image by $p$ of the canonical basis of $K n +1$ (consisting of the elements having only one nonzero entry, which is equal to 1), and $(1, 1, ..., 1)$ . On this basis, the homogeneous coordinates of $p (v)$ are simply the entries (coefficients) of $v$ .

Given another projective space $P (V)$ of the same dimension $n$ , and a frame $F$ of it, there is exactly one homography $h$ mapping $F$ onto the canonical frame of $P (K n +1)$ . The projective coordinates of a point $a$ on the frame $F$ are the homogeneous coordinates of $h (a)$ on the canonical frame of $P n (K)$ .

In the case of a projective line, a frame consists of three distinct points. If $P 1 (K)$ is identified with $K$ with a point at infinity $\infty$ added, then its canonical frame is $(\infty, 0, 1)$ . Given any frame $(a 0, a 1, a 2$ ), the projective coordinates of a point $a \neq a 0$ are $(r, 1)$ , where $r$ is the cross-ratio $(a, a 2; a 1, a 0)$ . If $a = a 0$ , the cross ratio is the infinity, and the projective coordinates are $(1,0)$ .

Related Research Articles

Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane. It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier Euclidean is used to distinguish it from other spaces that were later discovered in physics and modern mathematics.

In mathematics, a set $B$ of vectors in a vector space $V$ is called a basis if every element of $V$ may be written in a unique way as a finite linear combination of elements of $B$ . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to $B$ . The elements of a basis are called basis vectors.

In Euclidean geometry, an affine transformation, or an affinity, is a geometric transformation that preserves lines and parallelism.

Projective space Completion of the usual space with "points at infinity"

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.

In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.

In mathematics, a quadratic form is a polynomial with terms all of degree two. For example,

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space $over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .$

In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point.

In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.

In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex. The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex.

In mathematics, the Plücker map embeds the Grassmannian $, whose elements are k -dimensional subspaces of an n -dimensional vector space V, in a projective space, thereby realizing it as an algebraic variety. More precisely, the Plücker map embeds into the projectivization of the -th exterior power of . The image is algebraic, consisting of the intersection of a number of quadrics defined by the Plücker relations .$

In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space.

In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, $is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function.$

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by $, is the factor by which the eigenvector is scaled.$

In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra $Λ(V)$ of a vector space $V$ . This algebra is graded, associative and alternating, and consists of linear combinations of simple $k$ -vectors of the form

In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.

In projective geometry, a collineation is a one-to-one and onto map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism. The set of all collineations of a space to itself form a group, called the collineation group.

In geometry, a real point is a point in the complex projective plane with homogeneous coordinates $(x, y, z)$ for which there exists a nonzero complex number $λ$ such that $λx$ , $λy$ , and $λz$ are all real numbers.

In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V is finite dimensional and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V.

References

↑ Baer 2005 , p. 66
↑ Berger 2009 , chapter 6

Baer, Reinhold (2005) [First published 1952], Linear Algebra and Projective Geometry, Dover, ISBN 9780486445656
Berger, Marcel (2009), Geometry I, Springer-Verlag, ISBN 978-3-540-11658-5 , translated from the 1977 French original by M. Cole and S. Levy, fourth printing of the 1987 English translation

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Baer 2005 , p. 66

[2] Berger 2009 , chapter 6

[1]

[2]