for the computing company Quadrics In mathematics, a quadric , or quadric surface , is any D dimensional hypersurface in D 1 dimensional space defined as the locus mathematics locus of root of a function zeros of a quadratic polynomial . In coordinates nowrap x sub 1 sub , x sub 2 sub , ..., x sub D 1 sub , the general quadric is defined by the algebraic equation ref name geom http www.geom.uiuc.edu docs reference CRC formulas node61.html geom.uiuc.edu , Quadrics in Geometry Formulas and Facts by Silvio Levy, excerpted from 30th Edition of the CRC Standard Mathematical Tables and Formulas CRC Press . ref math sum i,j 1 D 1 x i Q ij x j sum i 1 D 1 P i x i R 0 math which may be compactly ... taken to be real number s or complex number s, but in fact, a quadric may be defined over any ... variety , and is studied in the branch of algebraic geometry . A quadric is thus an example of an algebraic variety. For the projective theory see quadric projective geometry . Euclidean plane and space ... as quadric surfaces . By making a suitable Euclidean change of variables, any quadric in Euclidean ... axis theorem principal axes of the quadric. In three dimensional Euclidean space there are 16 such normal ... white margin 1em auto 1em auto colspan 3 style background color white Non degenerate real quadric surfaces ... Of Two Sheets Quadric.png 150px colspan 3 style background color white Degenerate quadric surfaces ..., every quadric is defined by an equation of the form math Q X sum ij a ij X iX j 0 , math where ... in projective space exhibits the quadric as a projective algebraic variety . The quadric is said ... Focus geometry , an overview of properties of conic sections related to the foci. Klein quadric Quadratic function Superquadrics References reflist springer id q q076220 title Quadric first V.A. last Iskovskikh mathworld urlname Quadric title Quadric External links http www.professores.uff.br hjbortol arquivo 2007.1 qs quadric surfaces en.html Interactive Java 3D models of all quadric surfaces ... more details
for the Klein quartic Klein quartic The lines of a 3 dimensional projective space , S , can be viewed as points of a 5 dimensional projective space, T . In that 5 space, the points that represent a line in S lie on a hyperbolic quadric , Q known as the Klein quadric . If the underlying vector space of S is the 4 dimensional vector space V , then T has as underlying vector space the 6 dimensional exterior square sup 2 sup V of V . The line coordinates obtained this way are known as Pl cker coordinates . These Pl cker coordinates satisfy the quadratic relation math p 12 p 34 p 13 p 42 p 14 p 23 0 math defining Q , where math p ij u i v j u j v i math are the coordinates of the line spanned by the two vectors u and v . The 3 space, S , can be reconstructed again from the quadric, Q the planes contained in Q fall into two equivalence classes , where planes in the same class meet in a point, and planes in different classes meet in a line or in the empty set. Let these classes be math C math and math C math . The geometry of S is retrieved as follows The points of S are the planes in C . The lines of S are the points of Q . The planes of S are the planes in C . The fact that the geometries of S and Q are isomorphic can be explained by the isomorphism of the Dynkin diagram s A sub 3 sub and D sub 3 sub . References citation title Twistor Geometry and Field Theory first1 Richard Samuel last1 Ward first2 Raymond O Neil, Jr. last2 Wells publisher Cambridge University Press year 1991 isbn 978 0521422680 . geometry stub Category Projective geometry Category Quadrics ... more details
Unreferenced date December 2009 In projective geometry a quadric is the set of points of a projective space where a certain quadratic form on the homogeneous coordinates becomes zero. It may also be defined ... the set of all nonzero multiples of math v math . The projective quadric defined by math F math is the set ... projective plane , the quadric is also called a projective quadratic curve , conic conic section ... , the quadric is also called a projective quadratic surface . In general, if math K math is the field of real numbers, a quadric is an math n 1 math dimensional sub manifold of the projective space math ... vector math v math , the quadric consists of all points of math P math if math F math is a definite quadratic form definite form everywhere positive, or everywhere negative , the quadric is empty if math F math factors into the product of two non trivial linear form s, the quadric is the union of two hyperplanes and so on. Some authors may define quadric so as to exclude some or all of these special ... elements math M i i math determine whether the quadric is degenerate or not. We should give the analysis here. Polarity, tangent hyperplane, and singular points In general, a projective quadric ... ast math , relative to the chosen basis of math V math , is math F v math . If math p math is not on the quadric ... not contain math p math . If math p math is on the quadric and the hyperplane math h p ast math is well ... that is tangent space tangent to the quadric at math p math . If math p math is on the quadric ... is not defined, and math p math is said to be a singular point or singularity of the quadric. The tangent ... v 0 math . The quadric has singular points if and only the matrix math M math , in diagonal form, has one or more zeros in its diagonal. It follows that the set of all singular points on the quadric ... intersect a quadric math Q math at zero, one, or two points, or may be entirely contained in it. The line ... on whether math Delta B 2 4 A C math is negative, zero, or positive, respectively. DEFAULTSORT Quadric ... more details
Wikify date March 2010 In mathematics , an ovoid O of a finite polar space of rank r is a set of points, such that every subspace of rank math r 1 math intersects O in exactly one point. Cases Symplectic polar space An ovoid of math W 2 n 1 q math a symplectic polar space of rank n would contain math q n 1 math points. However it only has an ovoid if and only math n 2 math and q is even. In that case, when the polar space is embedded into math PG 3,q math the classical way, it is also an ovoid in the projective geometry sense. Hermitian polar space Ovoids of math H 2n,q 2 n geq 2 math and math H 2n 1,q 2 n geq 1 math would contain math q 2n 1 1 math points. Hyperbolic quadrics An ovoid of a hyperbolic quadric math Q 2n 1,q n geq 2 math would contain math q n 1 1 math points. Parabolic quadrics An ovoid of a parabolic quadric math Q 2 n,q n geq 2 math would contain math q n 1 math points. For math n 2 math , it is easy to see to obtain an ovoid by cutting the parabolic quadric with a hyperplane, such that the intersection is an elliptic quadric. The intersection is an ovoid. If q is even, math Q 2n,q math is isomorphic as polar space with math W 2 n 1 q math , and thus due to the above, it has no ovoid for math n geq 3 math . Elliptic quadrics An ovoid of an elliptic quadric math Q 2n 1,q n geq 2 math would contain math q n 1 math points. See also Ovoid projective geometry Category Incidence geometry geometry stub ... more details
For the hypersurface parameterizing lines in 3 space, also sometimes called a Pl cker surface Pl cker coordinates In algebraic geometry , a Pl cker surface , studied by harvs txt authorlink Julius Pl cker first Julius last Pl cker year 1899 , is a quartic surface in 3 dimensional projective space with a double line and 8 nodes. Construction For any quadric line complex , the lines of the complex in a plane envelop a quadric in the plane. A Pl cker surface depends on the choice of a quadric line complex and a line, and consists of points of the quadrics associated to the planes through the chosen line. References Citation authorlink R. W. H. T. Hudson last1 Hudson first1 R. W. H. T. title Kummer s quartic surface publisher Cambridge University Press series Cambridge Mathematical Library isbn 978 0 521 39790 2 mr 1097176 year 1990 url http www.archive.org details 184605691 Citation last1 Jessop first1 C. M. title Quartic surfaces with singular points url http digital.library.cornell.edu cgi t text text idx?c math idno 04290002 publisher Cornell University Library isbn 978 1 4297 0393 2 year 1916 Citation last1 Miles first1 Henry J. title On a Generalization of Plucker s Surface jstor 1968230 publisher Annals of Mathematics series Second Series year 1930 journal Annals of Mathematics issn 0003 486X volume 31 issue 3 pages 355 365 doi 10.2307 1968230 Citation last1 Pl cker first1 Julius title Neue geometrie des raumes gegr ndet auf die betrachtung der geraden linie als raumelement. url http www.archive.org details neuegeoraum00plucrich publisher University of Michigan Library isbn 978 1 4181 6773 8 year 1899 DEFAULTSORT Plucker Surface Category Algebraic surfaces sl Pl ckerjeva ploskev ... more details
see also Geometric algebra Grassmann Cayley algebra , also known as double algebra , is a form of modeling algebra for use in projective geometry . The technique is based on work by German mathematician Hermann Grassmann on exterior algebra , and subsequently by British mathematician Arthur Cayley s work on matrix mathematics matrices and linear algebra . The technique uses subspace s as basic elements of computation, a formalism which allows the translation of synthetic geometric statements into Invariant mathematics invariant algebraic statements. This can create a useful framework for the modeling of conic s and quadric s among other forms, and in tensor mathematics. It also has a number of applications in robotics , particularly for the kinesthetic analysis of manipulators. External links http www.science.uva.nl ga faq.html Geometric Algebra FAQ http www.inria.fr rrrt rr 2665.html Uses of the technique Category Multilinear algebra Linear algebra stub Geometry stub ... more details
turns out to be a manifold known as the Lie quadric a quadric hypersurface in projective space . Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve ..., Lie noticed a remarkable similarity between the Lie quadric of spheres in 3 dimensions, and the space of lines in 3 dimensional projective space, which is also a quadric hypersurface in a 5 dimensional projective space, called the Pl cker or Klein quadric . This similarity led Lie to his famous line ... called cycles or Lie cycles. It turns out that they form a quadric hypersurface in a projective space of dimension 4 or 5, which is known as the Lie quadric. The natural symmetry mathematics symmetries of this quadric form a transformation group group of transformations known as the Lie transformations ... quadric, not of the plane sphere plus point at infinity. The point preserving transformations are precisely ... point at infinity, namely the affine conformal maps. Lie sphere geometry in the plane The Lie quadric The Lie quadric of the plane is defined as follows. Let R sup 3,2 sup denote the space R sup 5 ... of the Lie quadric. The projective space R P sup 4 sup is the space of lines through the origin ..., where x x sub 0 sub , x sub 1 sub , x sub 2 sub , x sub 3 sub , x sub 4 sub . The planar Lie quadric ... using the point 1,0,0,0,0 &isin R P sup 4 sup . Any point in the Lie quadric Q can then be represented ... v &lambda sup 2 sup 0. The orthogonal space to 1,0,0,0,0 , intersected with the Lie quadric, is the two ... 0 and v 0,0,0,0,1 &minus 1. Hence points x &lambda 1,0,0,0,0 v on the Lie quadric with &lambda 0 correspond ... 1,0,0,0,0 1,0,0,0,0 induces an Involution mathematics involution &rho of the Lie quadric which reverses ... there is a one to one correspondence between points on the Lie quadric and cycles in the plane ... quadric Q . By the incidence of cycles, a solution to the Apollonian problem compatible with the chosen ... actually lies in the Lie quadric, and any point q on this line defines a cycle incident with x ... more details
In algebraic geometry , a Segre surface , studied by harvs txt authorlink Corrado Segre first Corrado last Segre year 1884 and harvs txt authorlink Beniamino Segre first Beniamino last Segre year 1951 , is an intersection of two quadric surface quadrics in 4 dimensional projective space . They are rational surface s isomorphic to a projective plane blowing up blown up in 5 points with no 3 on a line, and are del Pezzo surface s of degree 4, and have 16 rational lines. The term Segre surface is also occasionally used for various other surfaces, such as a quadric in 3 dimensional projective space, or the hypersurface math x 1 x 2 x 3 x 2 x 3 x 4 x 3 x 4 x 5 x 4 x 5 x 1 x 5 x 1 x 2 0. , math References Citation last1 Segre first1 Corrado title Etude des diff rentes surfaces du 4 sup e sup ordre conique double ou cuspidale g n rale ou d compos e consid r es comme des projections de l intersection de deux vari t s quadratiques de l espace quatre dimensions url http dx.doi.org 10.1007 BF01443412 publisher Springer Berlin Heidelberg year 1884 journal Mathematische Annalen issn 0025 5831 volume 24 pages 313 444 Citation doi 10.1093 qmath 2.1.216 last1 Segre first1 Beniamino title On the inflexional curve of an algebraic surface in S sub 4 sub id MathSciNet id 0044861 year 1951 journal The Quarterly Journal of Mathematics. Oxford. Second Series issn 0033 5606 volume 2 issue 1 pages 216 220 Category Algebraic surfaces Category Complex surfaces ... more details
This is a list of surface s , by Wikipedia page. See also List of algebraic surfaces , List of curves , Riemann surface . Minimal surface s Costa surface Catenoid Helicoid Riemann s surface Saddle tower Gyroid Catalan surface Enneper surface orientability Non orientable surfaces Klein bottle Real projective plane Cross cap Roman surface Boy s surface Quadric s Sphere Spheroid Oblate spheroid Cone geometry Ellipsoid Hyperboloid of one sheet Hyperboloid of two sheets hyperbolic paraboloid a ruled surface Paraboloid Pseudospherical surfaces Dini s surface Pseudosphere Algebraic surface s See the list of algebraic surfaces . Cayley cubic Barth sextic Clebsch cubic Monkey saddle saddle like surface for 3 legs. Torus Dupin cyclide inversion of a torus Whitney umbrella Miscellaneous surfaces right conoid a Ruled surface Category Mathematics related lists Surfaces Category Surfaces pt Anexo Lista de superf cies sl Seznam ploskev ... more details
In mathematics , in the field of geometry , a polar space of rank n n &ge 3 , or projective index n 1, consists of a set P , conventionally the set of points, together with certain subsets of P , called subspaces , that satisfy these axioms Every subspace, together with its own subspaces, is isomorphic with a projective space projective geometry PG d , q with 1 &le d &le n 1 and q a prime power. By definition, for each subspace the corresponding d is its dimension. The intersection of two subspaces is always a subspace. For each point p not in a subspace A of dimension of n 1, there is a unique subspace B of dimension n 1 such that A &cap B is n 2 dimensional. The points in A &cap B are exactly the points of A that are in a common subspace of dimension 1 with p . There are at least two disjoint subspaces of dimension n 1. A polar space of rank two is a generalized quadrangle . Finite polar spaces where P is a finite set are also studied as combinatorics combinatorial objects . Examples In PG d , q , with d odd and d &ge 3, the set of all points, with as subspaces the totally isotropic subspaces of an arbitrary Symplectic topology symplectic polarity, forms a polar space of rank d 1 2. Let Q be a nonsingular quadric in PG n , q with character &omega . Then the index of Q will be g n w 3 2. The set of all points on the quadric, together with the subspaces on the quadric, forms a polar space of rank g 1. Let H be a nonsingular Hermitian variety in PG n , q sup 2 sup . The index of H will be math left lfloor frac n 1 2 right rfloor , math . The points on H , together with the subspaces on it, form a polar space of rank math left lfloor frac n 1 2 right rfloor , math . Classification Jacques Tits proved that a finite polar space of rank at least three, is always isomorphic with one of the three structures given above. This leaves only the problem of classifying generalized quadrangles. References Citation last1 Cameron ... more details
quote box align right width 33 quote The question recently arose in conversation whether a dissertation of 2 lines could deserve and get a Fellowship. ... Cayley s projective definition of length is a clear case if we may interpret 2 lines with reasonable latitude. ... With Cayley the importance of the idea is obvious at first sight. source harvtxt Littlewood 1986 pp 39 40 In mathematics, a Cayley Klein metric is a metric mathematics metric on the complement of a fixed quadric in projective space defined using a cross ratio . The first example was given by harvs txt last Cayley authorlink Arthur Cayley year 1859 , and they were studied further by harvs txt first Felix last Klein authorlink Felix Klein year1 1871 year2 1873 . The Cayley Klein metric can be used to define the distance in the Cayley Klein model of hyperbolic geometry . The Hilbert metric of a convex set is defined in a similar way. Definition Suppose that Q is a fixed quadric in projective space. If p and q are 2 points then the line through p and q intersects the quadric Q in 2 further points a and b . The Cayley Klein distance d p , q from p to q is proportional to the logarithm of the cross ratio math d p,q C log frac qa bp pa bq math for some fixed constant C . References Citation last1 Cayley first1 Arthur author1 link Arthur Cayley title A sixth memoir upon quantics jstor 108690 year 1859 journal Philosophical Transactions of the Royal Society of London issn 0080 4614 volume 149 pages 61 90 Citation last1 Klein first1 Felin title Ueber die sogenannte Nicht Euklidische Geometrie publisher Springer Berlin Heidelberg language German doi 10.1007 BF02100583 jfm 03.0231.02 year 1871 journal Mathematische Annalen issn 0025 5831 volume 4 pages 573 625 Citation last1 Klein first1 Felix title Ueber die sogenannte Nicht Euklidische Geometrie Zweiter Aufsatz publisher Springer Berlin Heidelberg doi 10.1007 BF01443189 jfm 05.0271.01 year 1873 journal Mathematische Annalen issn 0025 5831 volume 6 pages 112 145 ... more details
The OpenGL Utility Library GLU is a computer graphics library . It consists of a number of functions that use the base OpenGL library to provide higher level drawing routines from the more primitive routines that OpenGL provides. It is usually distributed with the base OpenGL package. Among these features are mapping between screen and world coordinates, generation of Texture mapping texture mipmap s, drawing of quadric surfaces, Nonuniform rational B spline NURBS , tessellation of polygonal primitives, interpretation of OpenGL error codes, an extended range of transformation routines for setting up viewing volumes and simple positioning of the camera, generally in more human friendly terms than the routines presented by OpenGL. It also provides additional primitives for use in OpenGL applications, including sphere s, cylinder geometry cylinder s and Disk mathematics disks . GLU functions can be easily recognized by looking at them because they all have code glu code as a prefix. An example function is code gluOrtho2D code which defines a two dimensional orthographic projection matrix. Specifications for GLU are available at the http www.opengl.org documentation specs OpenGL specification page See also OpenGL Utility Toolkit GLUT OpenGL User Interface Library GLUI freeglut Category OpenGL de OpenGL Utility Library es OpenGL Utility fr OpenGL utility library it OpenGL Utility Library pt GLU ru GLU ... more details
Quadric For example with m n 1 we get an embedding of the product of the projective line with itself in P sup 3 sup . The image is a quadric , and is easily seen to contain two one parameter families of lines. Over the complex number s this is a quite general non singular quadric. Letting math Z 0 Z 1 Z 2 Z 3 math be the homogeneous coordinates on P sup 3 sup , this quadric is given as the zero ... more details
refimprove date May 2010 In mathematics , the complex projective plane , usually denoted P sup 2 sup C , is the two dimensional complex projective space . It is a complex manifold described by three complex coordinates math z 1,z 2,z 3 in mathbf C 3, qquad z 1,z 2,z 3 neq 0,0,0 math where, however, the triples differing by an overall rescaling are identified math z 1,z 2,z 3 equiv lambda z 1, lambda z 2, lambda z 3 quad lambda in mathbf C , qquad lambda neq 0. math That is, these are homogeneous coordinates in the traditional sense of projective geometry . Topology The Betti number s of the complex projective plane are 1, 0, 1, 0, 1, 0, 0, ..... The middle dimension 2 is accounted for by the homology class of the complex projective line, or Riemann sphere , lying in the plane. The nontrivial homotopy groups of the complex projective plane are math pi 2 pi 5 mathbb Z math . The fundamental group is trivial and all other higher homotopy groups are those of the 5 sphere, i.e. torsion. Algebraic geometry In birational geometry , a complex rational surface is any algebraic surface birationally equivalent to the complex projective plane. It is known that any non singular rational variety is obtained from the plane by a sequence of blowing up transformations and their inverses blowing down of curves, which must be of a very particular type. As a special case, a non singular complex quadric in P sup 3 sup is obtained from the plane by blowing up two points to curves, and then blowing down the line through these two points the inverse of this transformation can be seen by taking a point P on the quadric Q , blowing it up, and projecting onto a general plane in P sup 3 sup by drawing lines through P . The group of birational automorphisms of the complex projective plane is the Cremona group . Differential geometry As a Riemannian manifold, the complex projective plane is a 4 dimensional manifold whose sectional curvature is quarter pinched. The rival normalisations are for th ... more details
In PG 3,q , with q a prime power greater than 2, an ovoid is a set of math q 2 1 math points, no three of which collinear the maximum size of such a set . ref more properly the term should be ovaloid and ovoid has a different definition which extends to projective spaces of higher dimension. However, in dimension 3 the two concepts are equivalent and the ovoid terminology is almost universally used, except most notably, in Hirschfeld. ref When math q 2 math the largest set of non collinear points has size eight and is the complement of a plane. ref harvnb Hirschfeld 1985 loc pg.33, Theorem 16.1.3 ref An important example of an ovoid in any finite projective three dimensional space are the math q 2 1 math points of an elliptic quadric all of which are projectively equivalent . When q is odd or math q 4 math , no ovoids exist other than the elliptic quadric s. ref harvnb Barlotti 1955 and harvnb Panella 1955 ref When math q 2 2 h 1 math another type of ovoid can be constructed the Jacques Tits Tits ovoid, also known as the Michio Suzuki Suzuki ovoid. It is conjectured that no other ovoids exist in PG 3,q . Through every point P on the ovoid, there are exactly math q 1 math tangents, and it can be proven that these lines are exactly the lines through P in one specific plane through P . This means that through every point P in the ovoid, there is a unique plane intersecting the ovoid in exactly one point. ref harvnb Hirschfeld 1985 loc pg. 34, Lemma 16.1.6 ref Also, if q is odd or math q 4 math every plane which is not a tangent plane meets the ovoid in a Conic section conic . ref harvnb Hirschfeld 1985 loc pg.35, Corollary ref See also Ovoid polar space Oval projective plane Inversive plane Notes reflist References citation last Barlotti first A. title Un estensione del teorema di Segre Kustaanheimo journal Boll. Un. Mat. Ital. year 1955 volume 10 pages 96 98 citation last Hirschfeld first J.W.P. title Finite Projective Spaces of Three Dimensions year 1985 publisher Ox ... more details
In mathematics, especially in algebraic geometry , a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface affine and projective. An affine quartic surface is the solution set of an equation of the form math f x,y,z 0 math where f is a polynomial of degree 4, such as f x , y , z x sup 4 sup y sup 4 sup xyz z sup 2 sup &minus 1. This is a surface in affine space . On the other hand, a projective quartic surface is a surface in projective space P sup 3 sup of the same form, but now f is a homogeneous polynomial of 4 variables of degree 4, so for example f x , y , z , w x sup 4 sup y sup 4 sup xyzw z sup 2 sup w sup 2 sup &minus w sup 4 sup . If the base field in R or C the surface is said to be real or complex . If on the other hand the base field is finite, then it is said to be an arithmetic quartic surface . Special quartic surfaces Dupin cyclide s The Fermat quartic , given by x sup 4 sup y sup 4 sup z sup 4 sup w sup 4 sup 0 an example of a K3 surface and tiled by 12 octagons, in the Dyck tiling named after Walther von Dyck . K3 surface s Klein quartic Kummer surface Pl cker surface Weddle surface See also Quadric surface The union of two quadric surfaces is a special case of a quartic surface Cubic surface The union of a cubic surface and a plane is another particular type of quartic surface References Citation authorlink R. W. H. T. Hudson last1 Hudson first1 R. W. H. T. title Kummer s quartic surface publisher Cambridge University Press series Cambridge Mathematical Library isbn 978 0 521 39790 2 id MathSciNet id 1097176 year 1990 url http www.archive.org details 184605691 Citation last1 Jessop first1 C. M. title Quartic surfaces with singular points url http digital.library.cornell.edu cgi t text text idx?c math idno 04290002 publisher Cornell University Library isbn 978 1 4297 0393 2 year 1916 geometry stub Category Complex surfaces Category Algebraic surfaces ... more details
extension of the notion of a quadric surface . Whereas a quadric can be described as the zero set ... quadric and cyclidic coordinate geometries. Many other cyclidic geometries can be obtained by studying ... more details
Following is a list of some mathematics mathematically well defined shape s. See also list of geometric shapes , list of polygons, polyhedra and polytopes , and list of curves . 0D with no surface Point geometry point 1D with 0D surface interval mathematics interval Line geometry line 2D with 1D surface B zier curve As Bt sup n sup 0 s 1 0 t 1 s t 1, A sup n sup , A sup n &minus 1 sup B , ..., B sup n sup are vectors circle x sup 2 sup y sup 2 sup r sup 2 sup ellipse parabola hyperbola Plane mathematics plane polygon chiliagon decagon enneagon googolgon hectagon heptagon hendecagon hexagon myriagon octagon pentagon quadrilateral triangle trapezium 3D with 1D surface helix x sin z y cos z 3D with 2D surface B zier triangle As Bt Cu sup n sup 0 s 1 0 t 1 0 u 1 s t u 1, A sup p sup B sup q sup C sup r sup vectors if p q r n and p , q , r are nonnegative integers cylinder geometry cylinder hyperplane m bius strip platonic solid dodecahedron hexahedron cube icosahedron octahedron tetrahedron torus doughnut quadric cone geometry cone cylinder geometry cylinder ellipsoid spheroid sphere hyperboloid paraboloid sphere 4D with 3D surface polychoron hecatonicosachoron hexacosichoron hexadecachoron icositetrachoron pentachoron simplex tesseract spherical cone 5D with 4D surfaces regular 5 polytopes 5 dimensional simplex 5 dimensional cross polytope 5 dimensional hypercube 5 measure polytope Fractal s Apollonian gasket Cantor set Dragon curve Koch snowflake L vy C curve Lyapunov fractal Mandelbrot set Sierpinski carpet Peano curve Sierpinski triangle See also List of mathematical topics Periodic table of shapes The Periodic table of mathematical shapes Category Mathematics related lists Shapes ... more details
Infobox software name DKBTrace title DKBTrace logo File screenshot File caption collapsible author David Kirk Buck, Aaron A. Collins developer released 1986 Start date 1986 MM DD df yes no discontinued latest release version 2.12 latest release date 1991 Start date and age 1991 MM DD df yes no latest preview version latest preview date Start date and age YYYY MM DD df yes no frequently updated DO NOT include this parameter unless you know what it does programming language operating system platform size language status genre Ray tracing graphics Ray tracing license website DKBTrace was a graphical Ray tracing graphics ray tracing program which was the forerunner of POV Ray . It had no Graphical user interface GUI and ran via the command line . It featured quadric shapes spheres, ellipsoids, planes, etc. , constructive solid geometry intersection, difference, union , and procedural textures like wood and marble. It was originally written for the Amiga by David Kirk Buck 1986 . Aaron A. Collins ported it to the PC and worked with David to produce subsequent versions. The last version DKBTrace 2.12 was built in 1991 and ran on Unix systems, PCs and Amigas. Version 2.12 was released to the POV Ray team to use as the basis of POV Ray . Some early POV Ray history written by David K. Buck is available at the POV Ray site http www.povray.org documentation view 3.6.0 7 The Early History of POV Ray http www.povray.org documentation view 3.6.0 8 The Original Creation Message Some of the old versions of DKBTrace are still available on the Aminet site. They were also released on disks 397, 513, and 514 of the famous Fish disks distributed by Fred Fish . External links ftp alfred.ccs.carleton.ca pub dkbtrace dkb2.12 DKBTrace 2.12 Category 3D graphics software Category Amiga raytracers Category 1986 software graphics software stub de DKBTrace ko DKBTrace ... more details
Hermitian varieties are in a sense a generalisation of quadric s, and occur naturally in the theory of polarities . Definition Let K be a field with an involutive automorphism math theta math . Let n be an integer math geq 1 math and V be an n 1 dimensional vectorspace over K . A Hermitian variety H in PG V is a set of points of which the representing vector lines consisting of isotropic points of a non trivial Hermitian sesquilinear form on V . Representation Let math e 0,e 1, ldots,e n math be a basis of V . If a point p in the projective space has homogenous coordinates math X 0, ldots,X n math with respect to this basis, it is on the Hermitian variety if and only if math sum i,j 0 n a ij X i X j theta 0 math where math a i j a j i theta math and not all math a ij 0 math If one construct the Hermitian matrix A with math A i j a i j math , the equation can be written in a compact way math X t A X theta 0 math where math X begin bmatrix X 0 X 1 vdots X n end bmatrix . math Tangent spaces and singularity Let p be a point on the Hermitian variety H . A line L through p is by definition tangent when it is contains only one point p itself of the variety or lies completely on the variety. One can prove that these lines form a subspace, either a hyperplane of the full space. In the latter case, the point is singular. Category Algebraic varieties Algebra stub ... more details
In algebraic geometry , a line complex is a 3 fold given by the intersection of the Grassmannian G 2, 4 embedded in projective space P sup 5 sup by Pl cker coordinates with a hypersurface. It is called a line complex because points of G 2, 4 correspond to lines in P sup 3 sup , so a line complex can be thought of as a 3 dimensional family of lines in P sup 3 sup . The linear line complex and quadric line complex are the cases when the hypersurface has degree 1 or 2 they are both rational varieties . References Citation last1 Griffiths first1 Phillip author1 link Phillip Griffiths last2 Harris first2 Joseph author2 link Joe Harris mathematician title Principles of algebraic geometry publisher John Wiley & Sons location New York series Wiley Classics Library isbn 978 0 471 05059 9 id MathSciNet id 1288523 year 1994 Citation last1 Jessop first1 C. M. title A treatise on the line complex origyear 1903 url http digital.library.cornell.edu cgi t text text idx?c math idno 06960001 publisher American Mathematical Society location Providence, R.I. isbn 978 0 8218 2913 4 id MathSciNet id 0247995 year 2001 Citation last1 Klein first1 Felix title Zur Theorie der Liniencomplexe des ersten und zweiten Grades publisher Springer Berlin Heidelberg doi 10.1007 BF01444020 year 1870 journal Mathematische Annalen issn 0025 5831 volume 2 issue 2 pages 198 226 Category Algebraic varieties Category Threefolds ... more details
Theodor Reye born 20 June 1838 in Ritzeb ttel , Germany and died 2 July 1919 in W rzburg , Germany was a Germans German mathematician . He contributed to geometry, particularly projective geometry and synthetic geometry . He is best known for his introduction of Configuration geometry configurations in second edition of his book, Geometrie der Lage Geometry of Position, 1876 . The Reye configuration of 12 points, 12 planes, and 16 lines is named after him. Reye also developed a novel solution to the following three dimensional extension of the problem of Apollonius Construct all possible spheres that are simultaneously tangent to four given spheres. ref cite book author Reye T year 1879 title Synthetische Geometrie der Kugeln publisher B. G. Teubner location Leipzig url http www.gutenberg.org files 17153 17153 pdf.pdf format PDF de icon ref Life Reye obtained his Ph.D. from the University of G ttingen in 1861. His dissertation was entitled Die mechanische W rme Theorie und das Spannungsgesetz der Gase The mechanical theory of heat and the potential law of gases . Mathematical work Reye worked on conic section s, quadric s and projective geometry . Reye s work on linear manifolds of projective plane pencils and of bundles on spheres influenced later work by Corrado Segre on manifolds. References references External links Theodor Reye 1892 http www.archive.org details diegeometrieder01reyegoog Die Geometrie der Lage from archive.org . http genealogy.math.ndsu.nodak.edu id.php?id 49394 Math Genealogy summary MacTutor id Reye title Theodor Reye Persondata Metadata see Wikipedia Persondata . NAME Reye, Theodor ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 20 June 1838 PLACE OF BIRTH DATE OF DEATH 2 July 1919 PLACE OF DEATH DEFAULTSORT Reye, Theodor Category Geometers Category German mathematicians Category 1838 births Category 1919 deaths de Theodor Reye ... more details
In mathematics, and especially differential geometry , an affine sphere is a hypersurface for which the Affine differential geometry The affine normal line affine normal s all intersect in a single point. ref name spring cite web url http eom.springer.de a a011110.htm publisher Springer Online References title Affine Sphere author E. V. Shikin ref The term affine sphere is used because they play an analogous role in affine differential geometry to that of ordinary spheres in Euclidean differential geometry. An affine sphere is called improper if all of the affine normals are constant. ref name spring In that case, the intersection point mentioned above lies on the hyperplane at infinity . Affine spheres have been the subject of much investigation, with many hundreds of research article s devoted to their study. ref cite web url http scholar.google.co.uk scholar?hl en&q 22affine sphere 22&btnG Search&as sdt 1 2C5&as ylo &as vis 0 title Google Scholar Search publisher Google Inc ref Examples All quadric s are affine spheres the quadrics that are also improper affine spheres are the paraboloid s. ref cite book last1 Buchin first1 S. title Affine differential geometry year 1983 publisher Sci. Press and Gordon & Breach location language isbn 0 677 31060 9 ref If is a smooth function on the plane and the determinant of the Hessian matrix is 1 then the graph of in three space is an improper affine sphere. ref cite journal last1 Ishikawa first1 G. last2 Machida first2 Y. year 2005 title Singularities of improper affine spheres and surfaces of constant Gaussian curvature arxiv math 0502154 ref References reflist DEFAULTSORT Affine sphere Category Differential geometry differential geometry stub ... more details
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