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Line with a point at infinity added
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
Projective_line
Projective line over the real numbers
of a real projective line are called projective transformations, homographies, or linear fractional transformations. They form the projective linear group
Real_projective_line
Completion of the usual space with "points at infinity"
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
Projective_space
Type of geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that
Projective_geometry
Construction in group theory
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
Projective_linear_group
Compact non-orientable two-dimensional manifold
real projective plane, denoted R P 2 {\displaystyle \mathbf {RP} ^{2}} or P 2 {\displaystyle \mathbb {P} _{2}} , is a two-dimensional projective space
Real_projective_plane
Isomorphism of projective spaces in geometry
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces
Homography
Geometric concept of a 2D space with "points at infinity" adjoined
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
Projective_plane
Concept in geometry
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
Point_at_infinity
Algebra where division is always defined
inspired by the topological picture ⊙ {\displaystyle \odot } of the real projective line together with an extra point ⊥ (bottom element) such that ⊥ = 0 / 0
Wheel_theory
Real numbers with an added point at infinity
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
Projectively extended real line
Projectively_extended_real_line
Projective construction in ring theory
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
Projective_line_over_a_ring
Mathematical concept
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
Complex_projective_space
Model of the extended complex plane plus a point at infinity
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
Riemann_sphere
Concept in projective geometry
duality and beyond that to duality in any finite-dimensional projective geometry. A projective plane C may be defined axiomatically as an incidence structure
Duality_(projective_geometry)
Coordinate system used in projective geometry
dimension of the projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three
Homogeneous_coordinates
Point found separated from another, given a point pair
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
Projective_harmonic_conjugate
Algebraic variety in a projective space
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
Projective_variety
Real numbers adjoined with a nil-squaring element
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
Dual_number
Block design in combinatorial mathematics
geometric language, they are projectivities of the projective line. They form a group under composition which is the projective special linear group PSL(2
Steiner_system
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
known as Hopf fibrations. First, one can replace the projective line by an n-dimensional projective space. Second, one can replace the complex numbers by
Hopf_fibration
Concept in mathematics
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates
Quaternionic_projective_space
Ruled surface over the projective line
the P 1 {\displaystyle \mathbb {P} ^{1}} -bundle (a projective bundle) over the projective line P 1 {\displaystyle \mathbb {P} ^{1}} , associated to
Hirzebruch_surface
Space with one dimension
{\displaystyle K} is a one-dimensional vector space over itself. The projective line over K , {\displaystyle K,} denoted P 1 ( K ) , {\displaystyle \mathbf
One-dimensional_space
Mathematical concept
line; i.e., use the isomorphism t − 1 ↔ u {\displaystyle t^{-1}\leftrightarrow u} , then the resulting scheme is, at least visually, the projective line
Gluing_schemes
Concept in geometry and topology
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
Line_at_infinity
2-dimensional complex projective space
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
Complex_projective_plane
Set of points in a projective line or conic
mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic
Projective_range
In projective geometry, points that define coordinates
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
Projective_frame
Invariant in projective geometry
is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry. The cross-ratio
Cross-ratio
Type of topological space
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
Real_projective_space
Mathematical object studied in the field of algebraic geometry
called a projective algebraic set if V = Z(S) for some S. An irreducible projective algebraic set is called a projective variety. Projective varieties
Algebraic_variety
Plane tiling corresponding to a polyhedron
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations
Projective_polyhedron
Surface containing a line through every point
being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry. In algebraic geometry, ruled surfaces
Ruled_surface
Locus of the zeros of a polynomial of degree two
affine algebraic set. Quadrics may also be defined in projective spaces; see § Normal form of projective quadrics, below. In coordinates x1, x2, ..., xD+1
Quadric
Rational function of the form (az + b)/(cz + d)
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
Möbius_transformation
Algebraic curve in projective 3-space
of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (the
Twisted_cubic
In projective geometry, a bijection between projective spaces that preserves collinearity
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
Collineation
Curve defined as zeros of polynomials
zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three
Algebraic_curve
2D surface which extends indefinitely
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
Plane_(mathematics)
Mathematical concept
compactification of the real numbers, which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity
Infinity
Subdivision of the plane by lines
considered in the projective plane rather than in the Euclidean plane, every two lines cross, and an arrangement is the projective dual to a finite set
Arrangement_of_lines
n-dimensional linear system of divisors on a line bundle on X. The choice of a projective embedding of X, modulo projective transformations is likewise equivalent
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
Geometry
osculation of curves, a manifestly projective-invariant topic, lack any comprehensive theory. The ideas of projective differential geometry recur in mathematics
Projective differential geometry
Projective_differential_geometry
In mathematics, a jumping line or exceptional line of a vector bundle over projective space is a projective line in projective space where the vector bundle
Jumping_line
Geometric system with a finite number of points
Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field (that is, the
Finite_geometry
in projective n-space Pn. It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For
Rational_normal_curve
Topics referred to by the same term
variety Projective linear group Projective module Projective line Projective object Projective transformation Projective hierarchy Projective connection
Projective
is developed in Hartshorne's Algebraic Geometry, for example. The projective line over Dedekind domain R {\displaystyle R} is a smooth, proper arithmetic
Arithmetic_surface
2021 science-fiction novel by Andy Weir
coinciding with the formation of a dim infrared line from the Sun to Venus (dubbed the Petrova line, after its first observer, Dr. Irina Petrova). As
Project_Hail_Mary
Concept in algebraic geometry
{\displaystyle X} into a projective space. A line bundle is ample if some positive power is very ample. An ample line bundle on a projective variety X {\displaystyle
Ample_line_bundle
Geometry with 7 points and 7 lines
for this plane, as a member of a family of projective spaces, is PG(2, 2). Here, PG stands for "projective geometry", the first parameter is the geometric
Fano_plane
Map in projective geometry
embedding is a map used in projective geometry to consider the cartesian product of two projective spaces as a projective variety. It is named after Corrado
Segre_embedding
Automorphism group of the Klein quartic
In mathematics, the projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra
PSL(2,7)
Geometry
Oriented projective geometry is an oriented version of real projective geometry. Whereas the real projective plane describes the set of all unoriented
Oriented_projective_geometry
Property of pairs of points in a cycle
mathematics, two pairs of points in a cyclic order such as the real projective line separate each other when they occur alternately in the order. Thus
Point-pair_separation
Circle-like pointset in a geometric plane
ovals which can not lie in any projective plane. In a projective plane a set Ω of points is called an oval, if: Any line l meets Ω in at most two points
Oval_(projective_plane)
Embedding a topological space into a compact space as a dense subset
that the projective plane RP2 is not the one-point compactification of the plane R2 since more than one point is added. Complex projective space CPn
Compactification (mathematics)
Compactification_(mathematics)
Classifies holomorphic vector bundles over the complex projective line
complex projective line. In particular every holomorphic vector bundle over C P 1 {\displaystyle \mathbb {CP} ^{1}} is a direct sum of holomorphic line bundles
Birkhoff–Grothendieck_theorem
Straight figure with zero width and depth
such as non-Euclidean, projective, and affine geometry. In the Greek deductive geometry of Euclid's Elements, a general line (now called a curve) is
Line_(geometry)
Washington Metro rapid transit line
The Silver Line is a rapid transit line of the Washington Metro system, consisting of 39 stations in Loudoun County, Fairfax County and Arlington County
Silver Line (Washington Metro)
Silver_Line_(Washington_Metro)
Geometric space whose points represent algebro-geometric objects of some fixed kind
{\displaystyle d} hypersurfaces of projective space P n {\displaystyle \mathbb {P} ^{n}} . This is given by the projective bundle H i l b d ( P n ) = P (
Moduli_space
Rational surface in 5-dimensional projective space
surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear
Veronese_surface
Coordinates used to specify position of a line
scalar then line represented remains the same. So (l, m, n) is a system of homogeneous coordinates for the line. If points in the real projective plane are
Line_coordinates
Möbius transformation generalized to rings other than the complex numbers
transformation is the restriction to the field of a projective transformation or homography of the projective line. When a, b, c, d are integers (or, more generally
Linear fractional transformation
Linear_fractional_transformation
Hesse, is a pair of points of the projective line canonically associated with a set of 3 points of the projective line. More generally, one can define the
Hessian_pair
Algebraic curve
with the curve C being defined as a ramified double cover of the projective line, the ramification occurring at the roots of f, and also for odd n at
Hyperelliptic_curve
Overview of and topical guide to geometry
infinity Projective line Projective plane Oval (projective plane) Roman surface Projective space Complex projective line Complex projective plane Fundamental
Outline_of_geometry
Study of complex manifolds and several complex variables
not in general affine or projective. By Serre's GAGA theorem, every projective complex analytic variety is actually a projective complex algebraic variety
Complex_geometry
Sphere-like surface
In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension d ≥ 3. Simple examples in a real projective space
Ovoid_(projective_geometry)
Type of mathematical curve
projective space of dimension three over the field of the complex numbers (or over an algebraic closure of k {\displaystyle k} ), whose projective
Plane_cubic_curve
Line formed by the real numbers
circle (namely, the real projective line), and the extra point can be thought of as an unsigned infinity. Alternatively, the real line has two ends, and the
Number_line
Family of geometric objects with a common property
in the unique projective extension of the affine plane to a projective plane a single point (point at infinity) is added to each line in the pencil of
Pencil_(geometry)
In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up
Semilinear_map
was not just a copy of the projective plane, but a copy of the projective plane together with an embedding into projective 5-space. Varieties were often
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Fiber bundle whose fibers are projective spaces
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a Pn-bundle
Projective_bundle
Optimal stable 2-systolic inequality
the areas of rational 2-cycles representing the class of the complex projective line C P 1 ⊂ C P n {\displaystyle \mathbb {CP} ^{1}\subset \mathbb {CP}
Gromov's inequality for complex projective space
Gromov's_inequality_for_complex_projective_space
Theorem
non-singular and projective. Suppose there is a morphism φ from S to the projective line, with general fibre also a projective line. Then the theorem
Noether's theorem on rationality for surfaces
Noether's_theorem_on_rationality_for_surfaces
Algebraic variety
one example of a complex algebraic manifold, since it is the complex projective line. Compact Riemann surfaces Riemann sphere Elliptic curves Grassmannian
Algebraic_manifold
Structure in algebraic geometry
The general hope is that equations like [projective line] = [line] + [point] [projective plane] = [plane] + [line] + [point] can be put on increasingly solid
Motive_(algebraic_geometry)
Special class of linear fractional transformations
In projective geometry, a special conformal transformation is a linear fractional transformation that is not an affine transformation. Thus the generation
Special conformal transformation
Special_conformal_transformation
modular symbols is spanned by symbols {α,β} for α, β in the rational projective line Q ∪ {∞} subject to the relations {α,β} + {β,γ} = {α,γ} Informally,
Modular_symbol
complex projective line is a projective R-variety. (In fact, it is a variety with Q as its minimal field of definition.) Viewing the real projective line as
Field_of_definition
of the distribution of values of a holomorphic curve in the complex projective line. Pseudoholomorphic curve Shiffman (1977), p.553 Min Ru (2001). Nevanlinna
Holomorphic_curve
Generalized Euclidean space in mathematics
v ] {\displaystyle [v]} are also referred to as rays or projective rays. Each such projective ray is a copy of the nonzero complex numbers, which is topologically
Projective_Hilbert_space
Branch of mathematics
form only in projective space. For these reasons, projective space plays a fundamental role in algebraic geometry. Nowadays, the projective space Pn of
Algebraic_geometry
the dual number projective line, which adjoins to the dual numbers a set of points at infinity. Topologically, this projective line is equivalent to
Laguerre_transformations
Map from algebra to geometric transforms
mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group P
Projective_representation
Subspace defined by a polynomial of degree 2 over a field
by working in projective space rather than affine space. An example is the quadric surface x y = z w {\displaystyle xy=zw} in projective space P 3 {\displaystyle
Quadric_(algebraic_geometry)
Point on a line segment which is equidistant from both endpoints
defined in projective geometry since there is no distinguished point to play the role of the point at infinity (any point in a projective range may be
Midpoint
Concept in algebraic geometry
morphisms are in fact projective; but, examples of proper varieties which are not projective can be found using toric geometry. Projective morphisms define
Morphism_of_schemes
Geometric theorem
that if the points of the projective line P1 are depicted by a rational normal curve in Pn, then the group of the projective transformations of Pn that
Hesse's_principle_of_transfer
Existence of a line through two points
existence of an ordinary line can also be posed for points in the real projective plane RP2 instead of the Euclidean plane. The projective plane can be formed
Sylvester–Gallai_theorem
Connects non-singular algebraic curves with compact Riemann surfaces
from a compact Riemann surface S {\displaystyle S} to the complex projective line P 1 ( C ) {\displaystyle \mathbb {P} ^{1}(\mathbb {C} )} ramified only
Belyi's_theorem
Concept in algebraic geometry
hypersurfaces in n-dimensional projective space is Fano if and only if the sum of their degrees is at most n. Weighted projective space P(a0,...,an) is a singular
Fano_variety
Aspect of mathematical group theory
structures of an abstract 6-element set as the projective line P1(F5) – the line has 6 points, and the projective linear group acts 3-transitively, so fixing
Automorphisms of the symmetric and alternating groups
Automorphisms_of_the_symmetric_and_alternating_groups
Unique point and line of a conic section
transformation of each point in the plane into its polar line and each line in the plane into its pole. In projective geometry, this affords a one-to-one correspondence
Pole_and_polar
Generalization of covers
a degree 2 branched covering of the corresponding projective elliptic curve to the projective line. The previous example may be generalized to any algebraic
Branched_covering
London Underground line
The Northern line is a London Underground line which runs between North London and South London. It is printed in black on the Tube map. It carries more
Northern_line
Euclidean space without distance and angles
spaces are contained in projective spaces. For example, an affine plane can be obtained from any projective plane by removing one line and all the points on
Affine_space
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