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Geometric concept of a 2D space with "points at infinity" adjoined
complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space. Not all projective planes can be
Projective_plane
Compact non-orientable two-dimensional manifold
considered to change under projective transformations. The name projective comes from perspective drawing: projecting an image from one plane onto another as viewed
Real_projective_plane
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
2-dimensional complex projective space
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
2D surface which extends indefinitely
complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space. Not all projective planes can be
Plane_(mathematics)
Concept in projective geometry
concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry. A projective plane C may
Duality_(projective_geometry)
Geometry with 7 points and 7 lines
standard notation 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
Fano_plane
smooth projective planes are special projective planes. The most prominent example of a smooth projective plane is the real projective plane E {\displaystyle
Smooth_projective_plane
Circle-like pointset in a geometric plane
In projective geometry an oval is a point set in a plane that is defined by incidence properties. The standard examples are the nondegenerate conics.
Oval_(projective_plane)
Curve defined as zeros of polynomials
algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous
Algebraic_curve
fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are
Fake_projective_plane
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
Relation between Lie algebras depicted as a square
Rosenfeld projective planes and notated as if they were projective planes. More broadly, these compact forms are the Rosenfeld elliptic projective planes, while
Freudenthal_magic_square
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
referred to as arcs in the literature, is the (k, d)-arcs. In a finite projective plane π (not necessarily Desarguesian) a set A of k (k ≥ 3) points such that
Arc_(projective_geometry)
Curve from a cone intersecting a plane
} . A projective mapping is a finite sequence of perspective mappings. As a projective mapping in a projective plane over a field (pappian plane) is uniquely
Conic_section
Type of projective plane
Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically a special type of translation plane. A translation plane is a projective
Moufang_plane
Mathematical concept
In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases
Plane_curve
Self-intersecting compact surface, an immersion of the real projective plane
In geometry, Boy's surface is an immersion of the real projective plane in three-dimensional space. It was discovered in 1901 by the German mathematician
Boy's_surface
Projective plane not satisfying Desargues' theorem
In mathematics, a non-Desarguesian plane is a projective plane (or sometimes an affine plane) that does not satisfy Desargues' theorem (named after Girard
Non-Desarguesian_plane
a projective plane. Since a projective plane is a self-dual configuration, the dual configuration of an affine plane is obtained from a projective plane
Truncated_projective_plane
Geometric system with a finite number of points
the projective planes, there are also seven lines; each point is on three lines, and each line contains three points. This particular projective plane is
Finite_geometry
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
Card game
an example of a finite projective plane. If there are 3 points in each line this creates a structure known as the Fano plane. This represents a simpler
Dobble
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
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
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
Unsolved problem in combinatorial geometry
of k lines. Variations of the problem consider the projective plane rather than the Euclidean plane, and require that the triangles not be crossed by any
Kobon_triangle_problem
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
Cubic_plane_curve
Projective plane
Cayley plane, lines and points may be defined in a natural way so that it becomes a 2-dimensional projective space, that is, a projective plane. It is
Cayley_plane
Projective line over the real numbers
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically
Real_projective_line
Axiomatically defined geometrical space
affine plane obtained by removing l from the plane Π. A projective plane with a translation line is called a translation plane and the affine plane obtained
Affine plane (incidence geometry)
Affine_plane_(incidence_geometry)
Partition of a sphere's surface into polygons
projective polyhedra (tessellations of the real projective plane) – just as the sphere has a 2-to-1 covering map of the projective plane, projective polyhedra
Spherical_polyhedron
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
Self-intersecting, highly symmetrical mapping of the real projective plane into 3D space
real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however
Roman_surface
Algebraic structure associated with a topological space
holes, but for example the real projective plane R P 2 {\displaystyle \mathbb {RP} ^{2}} and complex projective plane C P 2 {\displaystyle \mathbb {CP}
Homology_(mathematics)
Field of mathematics which studies incidence structures
more general setting of projective planes, but it still holds in the Euclidean plane. The theorem is: In a projective plane, every non-collinear set
Incidence_geometry
Non-orientable surface with one edge
Euclidean plane to the real projective plane by adding one more line, the line at infinity. By projective duality the space of lines in the projective plane is
Möbius_strip
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
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
Riemann_sphere
General concept and operation in mathematics
electric fields. In some projective planes, it is possible to find geometric transformations that map each point of the projective plane to a line, and each
Duality_(mathematics)
Concept in algebraic geometry
Pezzo surface is either a product of two projective lines (with d=8), or the blow-up of a projective plane in 9 − d points with no three collinear, no
Del_Pezzo_surface
Mathematical space with two coordinates
meaningfully compared, as they can in a more general symplectic surface. The projective plane does away with both distance and parallelism. A two-dimensional metric
Two-dimensional_space
Mathematical problem
result on the non-existence of finite projective planes is the Bruck–Ryser theorem, which says that if a projective plane of order n exists and n ≡ 1 (mod
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Notion in supervised machine learning
{C}}\,\Delta C_{0})=\operatorname {VCDim} ({\mathcal {C}})} A finite projective plane of order n is a collection of n2 + n + 1 sets (called "lines") over
Vapnik–Chervonenkis_dimension
Topological space that locally resembles Euclidean space
surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central
Manifold
Structure in combinatorial mathematics
The use of projective is due to P.Dembowski (Finite Geometries, Springer, 1968), in analogy with the most common example, projective planes, while square
Block_design
Element of an exterior algebra
Points in a real projective space Pn are defined to be lines through the origin of the vector space Rn+1. For example, the projective plane P2 is the set
Multivector
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
Concept in geometry
infinity, the plane at infinity or the hyperplane at infinity, in all cases a projective space of one less dimension. As a projective space over a field
Point_at_infinity
Method for specifying point positions
needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane without the use of infinity. In
Coordinate_system
Two-dimensional geometrical space
affine planes are not defined over a field, they will in general not be isomorphic. Two affine planes arising from the same non-Desarguesian projective plane
Affine_plane
Two-dimensional manifold
connected sum of two real projective planes, P # P, is the Klein bottle K. The connected sum of the real projective plane and the Klein bottle is homeomorphic
Surface_(topology)
Points and lines with equal incidences
In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines,
Configuration_(geometry)
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)
Algebraic curve in mathematics
in x.) It is usually understood that the curve is embedded in the projective plane, with the point O being the unique point at infinity. Many sources
Elliptic_curve
Plane algebraic curve defined by a 4th-degree polynomial
therefore a rational plane algebraic curve of genus zero. The cruciform curve has three double points in the real projective plane, at x=0 and y=0, x=0
Quartic_plane_curve
Cubic graph with 10 vertices and 15 edges
projective plane. This is the embedding given by the hemi-dodecahedron construction of the Petersen graph (shown in the figure). The projective plane
Petersen_graph
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
Branch of mathematics
plane, the sphere, and the torus, which can all be realized in three dimensions without self-intersection, and the Klein bottle and real projective plane
Topology
Theorem in projective geometry
for the real projective plane and for any projective space defined arithmetically from a field or division ring; that includes any projective space of dimension
Desargues's_theorem
Type of non-Euclidean geometry
absolute is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..." The discovery of hyperbolic geometry
Hyperbolic_geometry
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.) Second, the product of symmetric spaces is symmetric
Simple_Lie_group
Non-Euclidean geometry
the points of projective space. A notable property of the projective elliptic geometry is that for even dimensions, such as the plane, the geometry is
Elliptic_geometry
Type of topological space
Universal coefficient theorem. Complex projective space Quaternionic projective space Lens space Real projective plane See the table of Don Davis for a bibliography
Real_projective_space
Roughly, the number of k-dimensional holes on a topological surface
polynomial is 1 + x {\displaystyle 1+x\,} . The homology groups of the projective plane P are: H k ( P ) = { Z k = 0 Z 2 k = 1 { 0 } otherwise {\displaystyle
Betti_number
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
Concept in geometry and topology
geometry and topology, the line at infinity is a projective line that is added to the affine plane in order to give closure to, and remove the exceptional
Line_at_infinity
Polyhedron with regular congruent polygons as faces
projective polyhedra, and are the projective counterparts of the Platonic solids. The tetrahedron does not have a projective counterpart as it does not have
Regular_polyhedron
Shape
closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas of mathematics (projective geometry, technical
Oval
28 lines which touch a general quartic plane curve in two places
tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for which all 28 of these
Bitangents_of_a_quartic
Inequality in differential geometry
the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves contained in it. A student of
Pu's_inequality
Principle in geometry
and projective geometry, five points determine a conic (a degree-2 plane curve), just as two (distinct) points determine a line (a degree-1 plane curve)
Five_points_determine_a_conic
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
Topics referred to by the same term
called a plane. Euclidean plane geometry, is the most common meaning; it includes Plane analytic geometry Plane synthetic geometry Plane projective geometry
Plane geometry (disambiguation)
Plane_geometry_(disambiguation)
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
tangent to five given conics in the plane in general position. If the problem is considered in the complex projective plane CP2, the correct solution is 3264
Steiner's_conic_problem
One-dimensional complex manifold
admit complex structures but the Möbius strip, Klein bottle and real projective plane do not. Every compact Riemann surface is a complex algebraic curve
Riemann_surface
define a non-degenerate projective conic section in a projective plane over a field. The usual definition of a conic in projective space uses a quadratic
Steiner_conic
Hesse, is a pencil (one-dimensional family) of cubic plane curves in the complex projective plane, defined by the equation x 3 + y 3 + z 3 − λ x y z =
Hesse_pencil
Image plane located between an eye point and an object
drawing). G. B. Halsted included the picture plane in his book Synthetic Projective Geometry: "To 'project' from a fixed point M (the 'projection vertex')
Picture_plane
\end{cases}}} The Moulton plane is an affine plane in which Desargues' theorem does not hold. The associated projective plane is consequently non-Desarguesian
Moulton_plane
Geometrical structure
Minkowski plane. Because of the essential role of the circle (considered as the non-degenerate conic in a projective plane) and the plane description
Benz_plane
Set of hypergraph nodes to which every hyperedge is connected
r-uniform projective plane. The following projective planes are known to exist: H2: it is simply a triangle graph. H3: it is the Fano plane. Hp+1 exists
Vertex_cover_in_hypergraphs
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
Topics referred to by the same term
protocol based on the OBEX protocol Octonionic projective plane, or Cayley plane, a projective plane over the octonions Oriented polypropylene, and specifically
OPP
Geometry theorem
It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem"
Pappus's_hexagon_theorem
Mathematical idealization of the trace left by a moving point
plane curve is a curve for which X {\displaystyle X} is the Euclidean plane—these are the examples first encountered—or in some cases the projective plane
Curve
scheme is a scheme parametrizing sheaves on a projective scheme. More specifically, if X is a projective scheme over a Noetherian scheme S and if F is
Quot_scheme
Discrete dynamical system on polygons in the projective plane and on their moduli space
pentagram map is a discrete dynamical system acting on polygons in the projective plane. It defines a new polygon whose vertices are obtained as the intersection
Pentagram_map
Group realized geometrically by reflections across the sides of a triangle
group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in
Triangle_group
lattice). A similar result is given by Pu's inequality for the real projective plane from 1952, due to Pao Ming Pu, with an upper bound of π/2 for the systolic
Systoles_of_surfaces
Concept in algebraic geometry
arose first in the form of a linear system of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so
Linear_system_of_divisors
Topological invariant in mathematics
non-convex Kepler–Poinsot polyhedra. Projective polyhedra all have Euler characteristic 1, like the real projective plane, while the surfaces of toroidal polyhedra
Euler_characteristic
Family of geometric objects with a common property
divisors in which the parameter space is a projective line. Typical pencils of curves in the projective plane, for example, are written as λ C + μ C ′ =
Pencil_(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
Polyhedron with seven sides
tetrahemihexahedron, which can be seen as a tessellation of the real projective plane. No heptahedra are regular. There are 34 topologically distinct convex
Heptahedron
Statement about cubic curves in the projective plane
curves (plane curves of degree three) in the projective plane P2. The original form states: Assume that two cubics C1 and C2 in the projective plane meet
Cayley–Bacharach_theorem
Topics referred to by the same term
Look up projective in Wiktionary, the free dictionary. Projective may refer to Projective geometry Projective space Projective plane Projective variety
Projective
Symmetric arrangement of finite sets
finite projective plane; thus showing how finite geometry and combinatorics intersect. When q = 2, the projective plane is called the Fano plane. Combinatorial
Combinatorial_design
PROJECTIVE PLANE
PROJECTIVE PLANE
Girl/Female
German, Swedish
Protective Victory
Girl/Female
Irish
Protective.
Girl/Female
Indian
Protective Angel
Girl/Female
Irish
Protective.
Girl/Female
Celtic, French, German, Irish
Strong; Protective
Boy/Male
German
Protective
Boy/Male
Greek
Productive.
Girl/Female
Muslim/Islamic
Protective angel
Boy/Male
Polish
Protective shield.
Girl/Female
German, Italian, Swedish
Protective; Victorious Shield
Girl/Female
Muslim/Islamic
Protective angel
Girl/Female
German American
Protective.
Boy/Male
British, English, Netherlands
Protective
Boy/Male
Christian & English(British/American/Australian)
Protective Grace
Girl/Female
Muslim
Protective Angel
Boy/Male
Christian & English(British/American/Australian)
Protective Friend
Girl/Female
Indian
Protective Angel
Boy/Male
German
Protective
Girl/Female
Muslim
Protective Angel
Boy/Male
Arabic, Indian, Muslim, Sindhi
Protective; Safety
PROJECTIVE PLANE
PROJECTIVE PLANE
Boy/Male
Bengali, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Oriya, Sanskrit, Telugu
Sweet Odour Stone; A Jewel of Lord Vishnu
Girl/Female
Indian, Telugu
Love
Male
English
English occupational surname transferred to forename use, derived from Middle English calfhirde, CALVERT means "calf-herder."
Boy/Male
Tamil
Life giving
Surname or Lastname
English
English : from a pet form of the medieval personal name Jan (see Jayne).
Boy/Male
Tamil
Intelligent
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : occupational name for a maker of pots and pans, from an agent derivative of Middle English pail(e) (Old French paelle ‘frying pan’, ‘cooking pan’).
Surname or Lastname
Americanized spelling of German Deutsch.English
Americanized spelling of German Deutsch.English : ethnic name for a Dutchman, especially an immigrant Dutch weaver.
Boy/Male
Hindu, Indian
Bearing the Truth
Surname or Lastname
English
English : variant of Dear.
PROJECTIVE PLANE
PROJECTIVE PLANE
PROJECTIVE PLANE
PROJECTIVE PLANE
PROJECTIVE PLANE
n.
A part of mechanics which treats of the motion, range, time of flight, etc., of bodies thrown or driven through the air by an impelling force.
a.
Pertaining to projection, or to a projectile.
n.
The act of scheming or planning; also, that which is planned; contrivance; design; plan.
n.
A jutting out beyond a surface.
n.
The quality or state of projecting, or being projected; projection; protrusion.
n.
A jutting out; also, a part jutting out, as of a building; an extension beyond something else.
n.
The representation of something; delineation; plan; especially, the representation of any object on a perspective plane, or such a delineation as would result were the chief points of the object thrown forward upon the plane, each in the direction of a line drawn through it from a given point of sight, or central point; as, the projection of a sphere. The several kinds of projection differ according to the assumed point of sight and plane of projection in each.
n.
The scene before or around, in time or in space; view; prospect.
n.
Looking forward in time; acting with foresight; -- opposed to retrospective.
n.
A body projected, or impelled forward, by force; especially, a missile adapted to be shot from a firearm.
a.
Caused or imparted by impulse or projection; impelled forward; as, projectile motion.
a.
Bringing into being; causing to exist; producing; originative; as, an age productive of great men; a spirit productive of heroic achievements.
n.
Being within view or consideration, as a future event or contingency; relating to the future: expected; as, a prospective benefit.
n.
A perspective glass.
n.
The act of throwing or shooting forward.
a.
Having the quality or power of producing; yielding or furnishing results; as, productive soil; productive enterprises; productive labor, that which increases the number or amount of products.
n.
Any method of representing the surface of the earth upon a plane.
a.
Affording protection; sheltering; defensive.
n.
Of or pertaining to a prospect; furnishing a prospect; perspective.
a.
Projecting or impelling forward; as, a projectile force.