In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a point...

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...

of a real projective line are called projective transformations, homographies, or linear fractional transformations. They form the projective linear group...

mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring A (with 1), the projective line P1(A) over...

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that...

real projective plane, denoted ⁠ R P 2 {\displaystyle \mathbf {RP} ^{2}} ⁠ or ⁠ P 2 {\displaystyle \mathbb {P} _{2}} ⁠, is a two-dimensional projective space...

especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action...

In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces...

incidence properties of the resulting projective plane. The line at infinity is also called the ideal line. In projective geometry, any pair of lines always...

values are increasing and unbounded. The projectively extended real line may be identified with a real projective line in which three points have been assigned...

complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space...

the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space. Not all projective planes can...

inspired by the topological picture ⊙ {\displaystyle \odot } of the real projective line together with an extra point ⊥ (bottom element) such that ⊥ = 0 / 0...

or the hyperplane at infinity, in all cases a projective space of one less dimension. As a projective space over a field is a smooth algebraic variety...

manifolds. In projective geometry, the sphere is an example of a complex projective space and can be thought of as the complex projective line P 1 ( C ) {\displaystyle...

In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following...

duality and beyond that to duality in any finite-dimensional projective geometry. A projective plane C may be defined axiomatically as an incidence structure...

dimension of the projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three...

In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in...

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...

fibers over real projective space RPn with fiber S0. The Hopf construction gives circle bundles p : S2n+1 → CPn over complex projective space. This is actually...

geometric language, they are projectivities of the projective line. They form a group under composition which is the projective special linear group PSL(2...

In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates...

homogeneous coordinates in the traditional sense of projective geometry. The Betti numbers of the complex projective plane are 1, 0, 1, 0, 1, 0, 0, ..... The middle...

points of the projective line over D are equivalence classes in B under this relation: P(D) = B/~. They are represented with projective coordinates [a...

simply the Euler characteristic. By contrast, the projective linear group of the real projective line, PGL(2, R) need not fix any points – for example...

is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry. The cross-ratio...

being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry. In algebraic geometry, ruled surfaces...

the P 1 {\displaystyle \mathbb {P} ^{1}} -bundle (a projective bundle) over the projective line P 1 {\displaystyle \mathbb {P} ^{1}} , associated to...

standard round metric, the measure of projective space is exactly half the measure of the sphere. Real projective spaces are smooth manifolds. On Sn, in...