Projective geometry

type of geometry

model of the extended complex plane plus a point at infinity

thumb|Two intersecting planes: Two-dimensional planes are the hyperplanes in three-dimensional space.

In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids.

thumb|Points , , , and , , , are related by a projective transformation so their cross ratios, and are equal. In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points , , , on a line, their cross ratio is defined as

particular mapping that projects a sphere onto a plane

space of 1-dimensional linear subspaces (lines passing through the origin) in a vector space

geometric concept of a 2D space with a "point at infinity" adjoined

mathematics

geometrical representation of the space of pure and mixed states of a qubit

methods in computer graphics to project three-dimensional objects onto a plane by means of numerical calculations

limiting point in some geometric spaces

fractional linear transformation on the complex projective line

construction in geometry, which, given to a conic, associates a line (“polar”) to a point and a point (“pole”) to a line

map projection

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.

finite projective plane of order 2

a projection of a polytope into a figure of dimension smaller by one

point found separated from another, given a point pair

symmetry in projective geometry

In mathematics, a Grassmannian \mathbf{Gr}_k(V), also known as a Grassmann manifold, is a differentiable manifold that parameterizes the set of all k-dimensional linear subspaces of an n-dimensional vector space V over a field K that has a differentiable structure. For example, the Grassmannian \mathbf{Gr}_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf{P}(V) of one dimension lower than V. When V is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension k(n-k). In general they have the structure of a nonsin

family of geometric objects with a common property

one-dimensional projective space

four points that determine six distinct lines

method of assigning coordinates to every line in projective 3-space

two figures in a plane are perspective from a point O if the lines joining corresponding points of the figures all meet at O

construction in group theory

In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: \mbox{SL}(2,\mathbf{R}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \colon a,b,c,d \in \mathbf{R}\mbox{ and }ad-bc=1\right\}.

Kähler–Einstein metric on a complex projective space

algebraic variety defined within a projective space

binary relation in geometry

2-dimensional complex projective space

concept in geometry and topology

In mathematics, the projective special linear group , isomorphic to , is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements, PSL(2, 7) is the smallest nonabelian simple group after the alternating group A5 with 60 elements, isomorphic to .

closed immersion from a product of projective spaces (of dimension m and n) to a projective space (of dimension mn+m+n)

the space of lines in a complex vector space

In projective geometry, a collineation is a one-to-one and onto map (a bijection) 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.

nonlinear differential operator used to study conformal mappings

geometric structure

möbius transformation generalized to rings other than the complex numbers

short exact sequence of sheaves on projective space

homomorphism between modules, paired with the associated homomorphism between the respective base rings

curve created by a geometric operation

extension of the set of the real numbers by a point denoted ∞

in an 𝑛-dimensional projective space, an (𝑛+2)-tuple of points such that no hyperplane contains 𝑛+1 of them

type of topological space

algorithm to solve systems of equations in projective geometry

geometric curve associated with a quadrangle