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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 itself
Collineation
Isomorphism of projective spaces in geometry
is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of
Homography
Geometry with 7 points and 7 lines
the plane, the collineation group is doubly transitive meaning that any ordered pair of points can be mapped by at least one collineation to any other ordered
Fano_plane
Geometric concept of a 2D space with "points at infinity" adjoined
the collineations of PG(2, K) are compositions of homographies and automorphic collineations. Automorphic collineations are planar collineations. A projective
Projective_plane
Vector field
A matter collineation (sometimes matter symmetry and abbreviated to MC) is a vector field that satisfies the condition, L X T a b = 0 {\displaystyle {\mathcal
Matter_collineation
Construction in group theory
projective space. A related group is the collineation group, which is defined axiomatically. A collineation is an invertible (or more generally one-to-one)
Projective_linear_group
Vector field that preserves the Riemann tensor
A curvature collineation (often abbreviated to CC) is vector field which preserves the Riemann tensor in the sense that, L X R a b c d = 0 {\displaystyle
Curvature_collineation
is called a perspective collineation (central collineation in more modern terminology). Let φ be a perspective collineation of S2. Each point of the
Perspectivity
Type of vector field
An affine vector field (sometimes affine collineation or affine) is a projective vector field preserving geodesics and preserving the affine parameter
Affine_vector_field
German mathematician and astronomer (1790–1868)
Die Elemente der Mechanik des Himmels Barycentric coordinate system Collineation Homogeneous coordinates Möbius counter Möbius plane Wells, John C. (2008)
August_Ferdinand_Möbius
Completion of the usual space with "points at infinity"
collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations"
Projective_space
Concept in geometry
to a polarity of H, every central collineation of H0 extends to a central collineation of H, and the full collineation group of H has two point orbits (one
Hughes_plane
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
Affine vector field Curvature collineation Homothetic vector field Killing form Killing horizon Killing spinor Matter collineation Spacetime symmetries Thales
Killing_vector_field
Type of symmetry in physics
curvature collineations forms a Lie algebra under the Lie bracket operation (if the smoothness condition is dropped, the set of all curvature collineations need
Spacetime_symmetries
Geometric transformation that preserves lines but not angles nor the origin
least three elements, the first condition can be simplified to: f is a collineation, that is, it maps lines to lines. By the definition of an affine space
Affine_transformation
Vector field
In physics, a homothetic vector field (sometimes homothetic collineation or homothety) is a projective vector field which satisfies the condition: L X
Homothetic_vector_field
Property of points all lying on a single line
sets and so, are collineations. In projective geometry these linear mappings are called homographies and are just one type of collineation. In any triangle
Collinearity
Theorem about hexagons and conics
there exists a central collineation, which maps the one onto the other triangle. But only in special cases this collineation is an affine scaling. For
Brianchon's_theorem
Geometric system with a finite number of points
on the same line) to collinear points is called a collineation of the plane. The full collineation group is of order 168 and is isomorphic to the group
Finite_geometry
Branch of mathematics
motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines
Geometry
Projective plane not satisfying Desargues' theorem
of the then known existence results (for both collineation groups and planes having such a collineation group) in both the finite and infinite cases appears
Non-Desarguesian_plane
Algebraic curve in mathematics
{\displaystyle \mathbb {H} ^{2}} (generated by orientation-preserving collineations). Further, the orthogonal trajectories of these ellipses comprise the
Elliptic_curve
Sporadic simple group
Todd, J. A. (1966), "A representation of the Mathieu group M24 as a collineation group", Annali di Matematica Pura ed Applicata, Series 4, 71: 199–238
Mathieu_group_M24
Concept in projective geometry
be thought of as a collineation between a pair of specially related projective spaces and called a reciprocity. If this collineation is a projectivity
Duality_(projective_geometry)
Axiomatically defined geometrical space
plane corresponding to the Desarguesian plane of order nine since the collineation group of that projective plane acts transitively on the lines of the
Affine plane (incidence geometry)
Affine_plane_(incidence_geometry)
Concept in geometry
Form n + 1 {\displaystyle n+1} Name Notation Number of points Collineation group Alternating 2 r {\displaystyle 2r} Symplectic W ( 2 r − 1 , q ) {\displaystyle
Polar_space
Topics referred to by the same term
the CRC Reciprocal square root Reciprocity (projective geometry), a collineation from a projective space onto its dual space, taking points to hyperplanes
Reciprocity
Structure in combinatorial mathematics
1109/LCOMM.2012.042512.120457. S2CID 7586742. Aschbacher, Michael (1971). "On collineation groups of symmetric block designs". Journal of Combinatorial Theory.
Block_design
Group of 𝑛 × 𝑛 invertible matrices
contains PGL ( n , F ) {\displaystyle \operatorname {PGL} (n,F)} , is the collineation group of projective space, for n > 2 {\displaystyle n>2} , and thus semilinear
General_linear_group
alignment, ambilineal, ambilineality, bilinear, collinear, collinearity, collineation, curvilinear, curvilinearity, delineate, delineation, delineavit, line
List of Greek and Latin roots in English/L
List_of_Greek_and_Latin_roots_in_English/L
the collineation group. Specifically, E ( T , P ) {\displaystyle E(T,P)} is the conic at point P {\displaystyle P} afforded by the collineation T {\displaystyle
Steiner_conic
Class of quartic plane curves
the central Steiner conics in the hyperbolic plane produced by direct collineations; and each single-loop is the locus of points P {\displaystyle P} such
Cassini_oval
2-transvections. Classifying the intrinsic conics in the hyperbolic plane, using collineation invariants, he offered metric characterizations and highlighted a natural
John_Sarli
American mathematician (1885 - 1943)
1090/s0002-9947-1913-1500941-8. Mitchell, Howard H. (1913). "On some systems of collineation groups". Bull. Amer. Math. Soc. 20 (3): 134–138. doi:10.1090/s0002-9904-1913-02447-9
Howard_Hawks_Mitchell
delimit, limes, limit, limitation linea line- line align, collinear, collineation, linea, lineage, linear, linearity, multicollinearity lingua lingu- tongue
List of Latin words with English derivatives
List_of_Latin_words_with_English_derivatives
Danish-American mathematician
MR 1560023 Mitchell, Howard H. (1918), "Book Review: Finite Collineation Groups", Bulletin of the American Mathematical Society, 24 (5): 243–252
Hans_Frederick_Blichfeldt
American mathematician
titled "A Group-Theoretic Characterization of the General Projective Collineation Group", and summarized in the Proceedings of the National Academy of
Nathan_Mendelsohn
Theorem in projective geometry
usually take the form of assuming the existence of sufficiently many collineations of a certain type, which in turn leads to showing that the underlying
Desargues's_theorem
corresponds to the group direction preserving collineations of the 3-net. Pseudo-automorphisms correspond to collineations fixing the two axis of the coordinate
Isotopy_of_loops
Smallest 3D projective space
into two conjugacy classes of 120 under the action of PGL(4, 2) (the collineation group of the space); a correlation interchanges these two classes. It
PG(3,2)
set of a projective plane is called a collineation, if it maps lines onto lines. The continuous collineations of a compact projective plane P {\displaystyle
Smooth_projective_plane
Concept in mathematics
elements. Another example is provided by the subgroup of order 21 of the collineation group of the Fano plane generated by a 3-fold symmetry σ fixing a point
Frobenius_group
MR 0698347 Mitchell, Howard H. (1914), "Determination of All Primitive Collineation Groups in More than Four Variables which Contain Homologies", American
Mitchell's_group
ISSN 0013-8584, MR 2583779 Coxeter, Harold Scott MacDonald (1956), "The collineation groups of the finite affine and projective planes with four lines through
Hessian_group
Mathematical metric in geometry
group is obtained as the collineations for which the absolute is stable. Indeed, cross-ratio is invariant under any collineation, and the stable absolute
Cayley–Klein_metric
Vector field in conformal geometry
Curvature collineation Einstein manifold Homothetic vector field Invariant differential operator Killing vector field Matter collineation Spacetime symmetries
Conformal Killing vector field
Conformal_Killing_vector_field
Hungarian mathematician
generalizations over finite fields. One topic of his research is the collineation groups of ovals and embedding problems for arcs in ovals; these investigations
Gábor_Korchmáros
Locus of the zeros of a polynomial of degree two
{\mathcal {Q}}\cup {\mathcal {R}}\;} there exists an involutorial central collineation σ P {\displaystyle \sigma _{P}} with center P {\displaystyle P} and σ
Quadric
British geometer
Todd, J. A. (1966). "A representation of the Mathieu group M24 as a collineation group". Annali di Matematica Pura ed Applicata. 71 (4): 199–238. doi:10
J._A._Todd
alignment, ambilineal, ambilineality, bilinear, collinear, collinearity, collineation, curvilinear, curvilinearity, delineate, delineation, delineavit, line
List of Greek and Latin roots in English/H–O
List_of_Greek_and_Latin_roots_in_English/H–O
Austrian and British mathematician (1930–2000)
Wagner, A. (1959). "On projective and affine planes with transitive collineation groups". Mathematische Zeitschrift. 71: 186–199. doi:10.1007/BF01181398
Ascher_Wagner
Type of curve in hyperbolic geometry
an incidence geometry, the Steiner conic at a point P produced by a collineation T is the locus of intersections L ∩ T(L) for all lines L through P. This
Hypercycle_(geometry)
Circle-like pointset in a geometric plane
(1991) the collineation groups stabilizing each of these hyperovals have been determined. Note that in the original determination of the collineation group
Oval_(projective_plane)
Theorem in plane geometry
sense) by a given acute angle about a given center, is seen to be a collineation mapping the whole hyperbolic plane in a 1-1 way onto the inside of a
Hjelmslev's_theorem
Indian polymath (1907–1966)
classification of integers, Journal of the University of Bombay, 2, 18–20 1934 Collineations in path-space, Journal of the Indian Mathematical Society, 21, 68–72
Damodar_Dharmananda_Kosambi
Group of all affine transformations of an affine space
wrote: The set P {\displaystyle {\mathfrak {P}}} of all projective collineations of Pn is a group which we may call the projective group of Pn. If we
Affine_group
Swiss mathematician and educator (1843–1917)
and meteorology, Benteli published articles on applications of central collineation and on perspective. H. Flükiger, "Prof. Dr. Albert Benteli", in Mitteilungen
Albert_Benteli_(professor)
78-dimensional exceptional simple Lie group
outer automorphism group of order 2. The EIV form of E6 is the group of collineations (line-preserving transformations) of the octonionic projective plane
E6_(mathematics)
Generalized manifold
spherical building of SL3(F2) and the stabiliser can be identified with the collineation group of the Fano plane generated by a 3-fold symmetry σ fixing a point
Orbifold
(2011), "Sixteen-dimensional locally compact translation planes with collineation groups of dimension at least 38 {\displaystyle 38} ", Adv. Geom., 11
Topological_geometry
Set of n^3 + 1 points arranged into subsets of n + 1
(Desarguesian or not) such that the automorphism group Γ is induced by a collineation group of the plane. For q = 3, Grüning proved that a Ree unital can not
Unital_(geometry)
English mathematician (1923–1956)
Finite Primitive Collineation Groups which contain Homologies of Period Two, concerned the group-theoretic properties of collineations, geometric transformations
Christine_Hamill
German mathematician (1928–1971)
S2CID 122535748. Dembowski, Peter; Ostrom, T. G. (1968). "Planes of ordern with collineation groups of order n 2 ". Mathematische Zeitschrift. 103 (3): 239–258. doi:10
Peter_Dembowski
Special type of projective plane
represent a point, and l represent a line. A central collineation with center P and axis l is a collineation fixing every point on l and every line through
Translation_plane
correspondence. (Coolidge 1931, p. 126) collinear On the same line collineation A collineation is an isomorphism from one projective space to another, often
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
American mathematician (1871–1959)
well-known management consultant. Projective groups of perspective collineations in the plane treated synthetically. 1896. (PhD dissertation) Introduction
Arnold_Emch
American mathematician
Mathematical Society 11: 268–273. MR 1558211 1906: "The resolution of any collineation into perspective reflections", Transactions of the American Mathematical
Mellen_Woodman_Haskell
American mathematician (1869-1959)
types of projective transformations of the plane (1895); and Theory of collineations (1911). Mary Newson, as she now became, resigned her position at the
Mary_Frances_Winston_Newson
Application of Clifford algebra
dual space which are involved in non-trivial transformations known as collineations. Therefore, x {\displaystyle x} and x ⋆ {\displaystyle x\star } cannot
Plane-based_geometric_algebra
Swedish mathematician
that for n > 7, in less than n–2 dimensions, there are no groups of collineations that are isomorphic to the symmetric or alternating group on n symbols
Anders_Wiman
Automorphism group of the Klein quartic
automorphisms is instead the order 2 extension PGL(2, 7), and the group of collineations of the projective line is the complete symmetric group of the points
PSL(2,7)
particles in arbitrary spacetimes (1992) by A. Qadir and M. Sharif Matter collineations of spacetime homogeneous Gödel-type metrics (2003) by U. Camci and M
Muhammad_Sharif_(cosmologist)
Russian mathematician
Cohn-Vossen, Stephan (1938). "Die Kollineationen desn-dimensionalen Raumes" [Collineations of n-dimensional space]. Math. Ann. (in German). 115: 80–86. doi:10
Stefan_Cohn-Vossen
German mathematician (1898–1944)
Journal. First Series. 31: 388–419. (Theory and classification of the collineations by involution on the linear congruence of rays; habilitation thesis)
Gerhard_Haenzel
American mathematician (1881–1973)
Morley with the dissertation Some Invariants and Covariants of Ternary Collineations. From 1905 to 1907 he was an instructor at the University of Cincinnati
Henry_B._Phillips
Hessian configuration and its connection with the group of 360 plane collineations", Proceedings of the London Mathematical Society, Second Series, 4:
Grünbaum–Rigby_configuration
Affine space over the complex numbers
space P(A) as an algebraic variety is none other than the group of collineations PGL(F(A)). In contrast, the automorphism group of the affine space A
Complex_affine_space
1893 book on making polygons with origami
of the symmetries of the plane includes congruence, similarity, and collineations of the projective plane; this part of the book also covers some of the
Geometric Exercises in Paper Folding
Geometric_Exercises_in_Paper_Folding
Canadian mathematician and computer scientist
background Alma mater University of Wisconsin–Madison (PhD) Thesis Collineations of Projective Planes of Order 10 (1975) Doctoral advisor Richard Bruck
Sue_Whitesides
American mathematician
year later from the same institution. Her master's thesis was titled "Collineations of Space which Leave Invariant a Quadric Surface," and it built off
Helen_Brewster_Owens
Concept in projective geometry
correlations also transform lines into lines, so they may be considered to be collineations of the two spaces. In general n-dimensional projective space, a correlation
Correlation (projective geometry)
Correlation_(projective_geometry)
COLLINEATION
COLLINEATION
COLLINEATION
COLLINEATION
Boy/Male
Indian, Sanskrit
Without Any Self Interest; Selfless
Girl/Female
Hindu
Beautiful, Lovable, Assiduous, Successful
Boy/Male
Hindu, Indian
Intelligent; Brilliant; Successful Person who Study Excellent
Boy/Male
Hindu, Indian, Telugu
Truth; Happiness
Girl/Female
American, Australian
Ebony; Deeply Black Wood
Boy/Male
Hindu
Boy/Male
Indian, Sanskrit
Wearing; Land; Earth
Girl/Female
Spanish
Feminine of Armando.
Girl/Female
Muslim
Graceful
Boy/Male
Indian
Intellectual
COLLINEATION
COLLINEATION
COLLINEATION
COLLINEATION
COLLINEATION
n.
The act of aiming at, or directing in a line with, a fixed object.