wiktionary A hyperplane is a concept in geometry . It is a generalization of the plane geometry plane into a different number of dimensions. A hyperplane of an n dimensional space n dimensional space is a flat ... into two half space s. Technical description In geometry , a hyperplane of an n dimensional space ... space or a projective space , and the notion of hyperplane varies correspondingly in all cases however, any hyperplane can be given in coordinate s as the solution of a single due to the codimension ... geometry translation of a vector hyperplane . A hyperplane in a Euclidean space separates that space into two half space s, and defines a reflection mathematics reflection that fixes the hyperplane ... are described here. Affine hyperplanes An affine hyperplane is an affine space affine subspace of codimension 1 in an affine space . In Cartesian coordinates , such a hyperplane can be described ... of the hyperplane, and are given by the inequalities math a 1x 1 a 2x 2 cdots a nx n b math and math a 1x 1 a 2x 2 cdots a nx n b. math As an example, a line is a hyperplane in 2 dimensional space, and a plane is a hyperplane in 3 dimensional space. A line in 3 dimensional space is not a hyperplane ... hyperplane of a Euclidean space has exactly two unit normal vectors. Affine hyperplanes are used ... hyperplane is a linear subspace of codimension 1. Such a hyperplane is the solution of a single ... at infinity added. An affine hyperplane together with the associated points at infinity forms a projective hyperplane. One special case of a projective hyperplane is the infinite or ideal hyperplane , which is defined with the set of all points at infinity. In real projective space, a hyperplane does ... so that both sides of a lone hyperplane are connected to each other. See also hypersurface decision boundary ham sandwich theorem arrangement of hyperplanes separating hyperplane theorem supporting hyperplane theorem References references Charles W. Curtis 1968 Linear Algebra , page 62, Allyn & Bacon ... more details
Unreferenced date December 2009 In mathematics , a hyperplane section of a subset X of projective space P sup n sup is the intersection set theory intersection of X with some hyperplane H &mdash in other words we look at the subset X sub H sub of those elements x of X that satisfy the single linear condition L 0 defining H as a Euclidean subspace linear subspace . Here L or H can range over the dual projective space of non zero linear form s in the homogeneous coordinates , up to scalar multiplication . From a geometrical point of view, the most interesting case is when X is an algebraic subvariety &mdash for more general cases, in mathematical analysis , some analogue of the Radon transform applies. In algebraic geometry , assuming therefore that X is V , a subvariety not lying completely in any H , the hyperplane sections are algebraic set s with irreducible component s all of dimension n &minus 1. What more can be said is addressed by a collection of results known collectively as Bertini s theorem . The topology of hyperplane sections is studied in the topic of the Lefschetz hyperplane theorem and its refinements. Because the dimension drops by one in taking hyperplane sections, the process is potentially an inductive method for understanding varieties of higher dimension. A basic tool for that is the Lefschetz pencil . DEFAULTSORT Hyperplane Section Category Algebraic geometry ... more details
In mathematics , in particular projective geometry , the hyperplane at infinity , also called the ideal hyperplane , is an n &minus 1 dimensional projective space added to an n dimensional affine space A , such as the real affine n space math mathbb R n math , in order to obtain uniformity of incidence properties. Adding the points of this hyperplane called ideal points or points at infinity converts the affine space into an n dimensional projective space , such as the real projective space math mathbb R P n math . There is one ideal point added for each pair of opposite directions in A . By adding these ideal points, the entire affine space A is completed to a projective space P , which may be called the projective completion of A . Each affine subspace S of A is completed to a projective space projective subspace of P by adding to S all the ideal points corresponding to the directions of the lines contained in S . The resulting projective subspaces are often called affine subspaces of the projective space P , as opposed to the infinite or ideal subspaces, which are the subspaces of the hyperplane at infinity however, they are projective spaces, not affine spaces . In the projective space, each projective subspace of dimension k intersects the ideal hyperplane in a projective subspace at infinity whose dimension is k &minus 1. A pair of non parallel geometry parallel affine hyperplanes ... intersect at a projective subspace of the ideal hyperplane the intersection lies on the ideal hyperplane . Thus, parallel hyperplanes, which did not meet in the affine space, intersect in the projective completion due to the addition of the hyperplane at infinity. Similarly, parallel lines intersect ..., any hyperplane may be chosen to be the hyperplane at infinity. Specifically, if P is a projective space and H is a hyperplane of P , then P &minus H is an affine space whose projective completion is P . Thus, the ideal hyperplane cannot be identified in terms of P alone. See also Point at infinity ... more details
hyperplane, then math S math is a convex set. ref name Boyd The hyperplane in the theorem may not be unique ... in the third picture on the right. A related result is the separating hyperplane theorem . See also Image Supporting hyperplane3.svg right thumb A supporting hyperplane containing a given point on the boundary ... more details
Merge to Separating axis theorem date May 2010 In geometry , a maximum margin hyperplane is a hyperplane which separates two clouds of points and is at equal distance from the two. The margin between the hyperplane and the clouds is maximal. See the article on Support Vector Machines for more details. Category Euclidean geometry Geometry stub ... more details
In mathematics , specifically in algebraic geometry and algebraic topology , the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hyperplane section Y , the homology mathematics homology , cohomology , and homotopy group s of X determine those of Y . A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories. The Lefschetz hyperplane theorem for complex projective varieties Let X be an n dimensional complex projective algebraic variety in CP sup N sup , and let Y be a hyperplane section of X such that nowrap begin U X Y nowrap end is smooth. The Lefschetz theorem refers to any of the following statements ref Harvnb Milnor ... the hyperplane section Y alone, he put it into a family of hyperplane sections Y sub t sub , where nowrap begin Y Y sub 0 sub nowrap end . Because a generic hyperplane section is smooth, all but a finite ... an additional finite number of slits, the resulting family of hyperplane sections is topological ... , therefore, can be understood if one understands how hyperplane sections are identified across the slits ... hyperplane theorem for homotopy groups. An approach that does was found by Thom no later than ... found a generalization of the Lefschetz hyperplane theorem to the case where the coefficients ... in CP sup N sup . Then in the cohomology ring of X , the k fold product with the cohomology class of a hyperplane ... part of the Lefschetz hyperplane theorem. The hard Lefschetz theorem in fact holds for any compact ... vanishing second cohomology groups, so there is no analogue of the second cohomology class of a hyperplane ... Frankel first2 Theodore title The Lefschetz theorem on hyperplane sections id MathSciNet id 0177422 ... more details
This is a list of convexity topics , by Wikipedia page. Alpha blending Barycentric coordinates Borsuk s conjecture Bond convexity Carath odory s theorem convex hull Choquet theory Closed convex function concave function Concavity Convex analysis Convex combination Convex and concave Convex conjugate Convex function Convex geometry Convex hull Convex lens Convex optimization Convex polygon Convex set Epigraph mathematics Extreme point Fenchel conjugate Fenchel s inequality Fixed point theorems in infinite dimensional spaces Gift wrapping algorithm Graham scan Hadwiger conjecture combinatorial geometry Hadwiger s theorem Helly s theorem Hyperplane Indifference curve Infimal convolute Interval mathematics Jarvis march Jensen s inequality Lagrange multiplier Legendre transformation Locally convex topological vector space Mahler volume Minimal convex decomposition Minkowski s theorem Mixed volume Mixture density Newton polygon Proper convex function Radon s theorem Separating axis theorem Shapley Folkman lemma Shephard s problem Simplex Simplex method Subdifferential Supporting hyperplane Supporting hyperplane theorem Category Mathematics related lists Convexity Category Mathematical analysis Category Convex geometry Category Outlines ... more details
otheruses In geometry , a half space is either of the two parts into which a plane geometry plane divides the three dimensional euclidean space. More generally, a half space is either of the two parts into which a hyperplane divides an affine space . That is, the points that are not incident to the hyperplane are partition set theory partitioned into two convex set s i.e., half spaces , such that any subspace connecting a point in one set to a point in the other must intersect the hyperplane. One can have open and closed half spaces. An open half space is either of the two open set s produced by the subtraction of a hyperplane from the affine space. A closed half space is the union of an open half space and the hyperplane that defines it. If the space is two dimensional , then a half space is called a half plane open or closed . A half space in a one dimensional space is called a Line mathematics Ray ray . A half space may be specified by a linear inequality, derived from the linear equation that specifies the defining hyperplane. A strict linear inequality mathematics inequality math a 1x 1 a 2x 2 cdots a nx n b math specifies an open half space, while a non strict one math a 1x 1 a 2x 2 cdots a nx n geq b math specifies a closed half space. Here, one assumes that not all of the real numbers a sub 1 sub , a sub 2 sub , ..., a sub n sub are zero. Properties A half space is a convex set . Any convex set can be described as the possibly infinite intersection of half spaces. Upper and lower half spaces The open closed upper half space is the half space of all x sub 1 sub , x sub 2 sub , ..., x sub n sub such that x sub n sub 0 0 . The open closed lower half space is defined similarly, by requiring that x sub n sub be negative non positive . See also Upper half plane Poincar half plane model External links Mathworld urlname Half Space title Half Space DEFAULTSORT Half Space Category Euclidean geometry cs Poloprostor de Halbraum es Semiespacio fr Demi espace it Semispaz ... more details
In mathematics , the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane section s for smooth projective varieties over algebraically closed field s, introduced by Eugenio Bertini . This is the simplest and broadest of the Bertini theorems applying to a linear system of divisors simplest because there is no restriction on the characteristic of a field characteristic of the underlying field, while the extensions require characteristic 0. ref Springer id B b015770 title Bertini theorems ref ref Hartshorne, Ch. III.10. ref Statement for hyperplane sections of smooth varieties Let X be a smooth projective variety over an algebraically closed field. Let math H math denote the linear system of divisors complete system of hyperplane divisors in the projective space math mathbf P n math containing X . Recall that it is a variety isomorphic to the algebraic geometry of projective spaces projective space math mathbf P n math . The theorem of Bertini states that the smooth hyperplane sections not equal to X form a dense subset of the total system of divisors math H math . If dim X 2, then the hyperplane section is connected, hence irreducible. The theorem hence asserts that a general hyperplane section not equal to X is smooth, that is the property of smoothness is generic. Outline of a proof We consider the subfibration of the product variety math X times H math with fiber above math x in X math the linear system of hyperplanes that intersect X non transversality mathematics transversally at x . The rank of the fibration in the product is one less than the codimension of math X subset mathbf P n math , so that the total space has lesser dimension than math n math and so its projection is contained in a divisor of the complete system math H math . General statement Over math mathbb C math , a general element of a linear system of divisors is smooth away from the base locus of the system. Remarks The theorem of Bertini is often used ... more details
one source date June 2010 In geometric probability theory, Wendel s theorem , named after James G. Wendel, gives the probability that N points continuous uniform distribution distributed uniformly at random on an n sphere n dimensional hypersphere all lie on the same half of the hypersphere. In other words, one seeks the probability that there is some hyperplane intersecting the center of the hypersphere such that all the points lie on the same side of the hyperplane. Wendel s theorem says that the probability is ref citation last Wendel first J. G. title A Problem in Geometric Probability journal Math. Scand volume 11 year 1962 page 109&ndash 111 ref math p n,N 2 N 1 sum k 0 n 1 binom N 1 k . math References reflist Category Probability theorems Category Theorems in geometry ... more details
Unreferenced stub auto yes date December 2009 In a statistical classification statistical classification problem with two classes, a decision boundary or decision surface is a hypersurface that partitions the underlying vector space into two sets, one for each class. The classifier will classify all the points on one side of the decision boundary as belonging to one class and all those on the other side as belonging to the other class. If the decision surface is a hyperplane , then the classification problem is linear, and the classes are linearly separable . Decision boundaries are not always clear cut. That is, the transition from one class in the feature space to another is not discontinuous, but gradual. This effect is common in fuzzy logic based classification algorithms, where membership in one class or another is ambiguous. In ANNs and SVMs In the case of backpropagation based artificial neural network s or perceptron s, the type of decision boundary that the network can learn is determined by the number of hidden layers the network has. If it has no hidden layers, then it can only learn linear problems. If it has one hidden layer, then it can learn problems with convex decision boundaries and some concave decision boundaries . The network can learn more complex problems if it has two or more hidden layers. In particular, support vector machine s find a hyperplane that separates the feature space into two classes with the maximum margin hyperplane maximum margin . If the problem is not originally linearly separable, the kernel trick is used to turn it into a linearly separable one, by increasing the number of dimensions. Thus a general hypersurface in a small dimension space is turned into a hyperplane in a space with much larger dimensions. Neural networks try to learn the decision boundary which minimizes the empirical error, while support vector machines try to learn the decision boundary which gives the best generalization. DEFAULTSORT Decision Boundary C ... more details
Unreferenced date November 2006 A reciprocity is a collineation from a projective space onto its dual space , taking points to hyperplane s and vice versa and preserving incidence geometry incidence . If it can be represented as a homography , it is called a correlation projective geometry correlation . See also Reciprocity theorem DEFAULTSORT Reciprocity Projective Geometry Category Projective geometry Geometry stub ... more details
Orphan date February 2009 In mathematics, a supersoluble arrangement is a Arrangement of hyperplanes hyperplane arrangement which has a maximal Flag linear algebra flag with only modular elements. Examples include arrangements associated with Coxeter group s of type A and B. It is known that all Orlik Solomon algebra s of supersoluble arrangements are Koszul algebra s. ref Orlik Solomon Algebras in Algebra and Topology, S. Yuzvinsky ref References references Category Discrete geometry ... more details
planes of existence The physical plane actually hyperplane , physical world , or physical universe , in emanation ist metaphysics such as are found in Neoplatonism , Hermeticism , Hinduism and Theosophy , refers to the visible reality of space and time , energy and matter the physical universe in Occultism and esoteric cosmology is the lowest or density densest of a series of Plane cosmology planes of existence hyperplanes that are said to be nested . References Max Heindel Heindel, Max , The Rosicrucian Mysteries Chapter III http www.rosicrucian.com rms rmseng01.htm Chapter III The Visible and the Invisible Worlds , 1911, ISBN 0 911274 86 3 DEFAULTSORT Physical Plane Category Esoteric cosmology Category Paranormal worlds and bodies Category Shabd paths Category Theosophical philosophical concepts pt Plano f sico ... more details
about correlation in projective geometry correlation disambiguation Unreferenced date November 2006 A correlation is a duality projective geometry duality collineation from a projective space onto its dual space, taking points to hyperplane s and vice versa and preserving incidence geometry incidence from a projective space to itself. In the case of Projective plane projective planes correlations can only exist if the plane is self dual. If a correlation is Involution mathematics involutory that is, two applications of the correlation equals the identity P P for all points P then it is called a pole and polar polarity . DEFAULTSORT Correlation Projective Geometry Category Projective geometry Geometry stub ... more details
Planes of existence The monadic plane hyperplane or spacetime continuum universe , enclosing and interpenetrating grosser hyperplanes, respectively is the plane in which the monad or holy spirit or oversoul is said to exist. The term is from the Greek word monad and is used in the Arcane School ideas. Theosophical differences Classical 1800s Theosophy does not say the monad is human, but Annie Besant & Charles Leadbeater may have said so, and Alice Bailey or others, who uses similar ideas, did say so. See also Paranirvana , Anupadaka . References reflist DEFAULTSORT Monadic Plane Category Esoteric cosmology Category Paranormal worlds and bodies ... more details
Planes of existence The soul ful or to Indian neo Theosophists buddhic plane hyperplane or spacetime continuum universe , separately, or enclosing and interpenetrating grosser hyperplanes, respectively or world simply in Theosophy is the world in which buddhi , i.e. soul , exists. References reflist Sources div class references small Charles Leadbeater 1912 1937 , A Textbook of Theosophy , Madras, India Theosophical Publishing House, 1912 1937, div DEFAULTSORT Psychic Plane Category Esoteric cosmology Category Paranormal worlds and bodies Category Shabd paths Category Theosophical philosophical concepts ... more details
In convex analysis and mathematical optimization , the supporting functional is a generalization of the supporting hyperplane of a set. Mathematical definition Let X be a locally convex topological space , and math C subset X math be a convex set , then the continuous linear functional math phi X to mathbb R math is a supporting functional of C at the point math x 0 math if math phi x leq phi x 0 math for every math x in C math . ref cite book title Foundations of mathematical optimization convex analysis without linearity page 323 first1 Diethard last1 Pallaschke first2 Stefan last2 Rolewicz publisher Spring year 1997 isbn 9780792344247 ref Relation to support function If math h C X to mathbb R math where math X math is the dual space of math X math is a support function of the set C , then if math h C left x right x left x 0 right math , it follows that math h C math defines a supporting functional math phi X to mathbb R math of C at the point math x 0 math such that math phi x x x math for any math x in X math . Relation to supporting hyperplane If math phi math is a supporting functional of the convex set C at the point math x 0 in C math such that math phi left x 0 right sigma sup x in C phi x inf x in C phi x math then math H phi 1 sigma math defines a supporting hyperplane to C at math x 0 math . ref cite book last1 Borwein first1 Jonathan authorlink1 Jonathan Borwein last2 Lewis first2 Adrian title Convex Analysis and Nonlinear Optimization Theory and Examples edition 2 year 2006 publisher Springer isbn 9780387295701 page 240 ref References Reflist Category Functional analysis Category Duality theories Category Types of functions ... more details
Image Tile 4,4.svg thumb Square tiling four square geometry square faces per vertex Image hexahedron.png thumb Cube three square geometry square faces per vertex In geometry , a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square geometry square s that bound a cube is a face of the cube. The suffix hedron is derived from the Greek word hedra which means face . Sometimes, in the case of a pyramid , the term face is understood to exclude the base. The two dimensional polygons that bound higher dimensional polytopes are also commonly called faces . Formally, however, a face is any of the lower dimensional boundaries of the polytope, more specifically called an n face . Formal definition In convex geometry , a face of a polytope P is the intersection of any supporting hyperplane of P and P . From this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. For example, a polyhedron R sup 3 sup is entirely on one hyperplane of R sup 4 b . If R sup 4 sup were spacetime, the hyperplane at t 0 supports and contains the entire polyhedron. Thus, by the formal definition, the polyhedron is a face of itself. All of the following are the n faces of a 4 polytope 4 dimensional polytope 4 face the 4 dimensional 4 polytope itself 3 face any 3 dimensional cell geometry cell 2 face any 2 dimensional polygonal face using the common definition of face 1 face any 1 dimensional edge geometry edge 0 face any 0 dimensional vertex geometry vertex the empty set. Facets If the polytope lies in m dimensions, a face in the m 1 dimension is called a Facet mathematics facet . For example, a cell of a polychoron is a facet, a face of a polyhedron is a facet, an edge of a polygon is a facet, etc. A face in the n 2 dimension is called a Ridge geometry ridge . See also Euler characteristic External links GlossaryForHyperspace anchor Face title Face mathworld urlname Face title Face Category Elementary ... more details
In mathematics , a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz , in order to analyse the algebraic topology of an algebraic variety V . A pencil here is a particular kind of linear system of divisors on V , namely a one parameter family, parametrised by the projective line . This means that in the case of a complex algebraic variety V , a Lefschetz pencil is something like a fibration over the Riemann sphere but with two qualifications about singularity. The first point comes up if we assume that V is given as a projective variety , and the divisors on V are hyperplane section s. Suppose given hyperplanes H and H &prime , spanning the pencil &mdash in other words, H is given by L 0 and H &prime by L &prime 0 for linear forms L and L &prime , and the general hyperplane section is V intersected with math lambda L mu L prime 0. math Then the intersection J of H with H &prime has codimension two. There is a rational mapping math V rightarrow P 1 math which is in fact well defined only outside the points on the intersection of J with V . To make a well defined mapping, some blowing up must be applied to V . The second point is that the fibers may themselves degenerate and acquire Mathematical singularity singular points where Bertini s lemma applies, the general hyperplane section will be smooth . A Lefschetz pencil restricts the nature of the acquired singularities, so that the topology may be analysed by the vanishing cycle method. The fibres with singularities are required to have a unique quadratic singularity, only. ref Springer id m m064700 title Monodromy transformation ref It has been shown that Lefschetz pencils exist in characteristic zero . They apply in ways similar to, but more complicated than, Morse function s on smooth manifold s. Simon Donaldson has found a role for Lefschetz pencils in symplectic topology , leading to more recent research interest in them. References S. K. Donaldson, Lefschetz Fibrations in S ... more details
Unreferenced date December 2009 In mathematics , complex dimension usually refers to the dimension of a complex manifold M , or complex algebraic variety V . If the complex dimension is d , the real dimension will be 2 d . That is, the smooth manifold M has dimension 2 d and away from any Mathematical singularity singular points V will also be a smooth manifold of dimension 2 d . The same points apply to codimension . For example a smooth complex hypersurface in complex projective space of dimension n will be a manifold of dimension 2 n &minus 1 . A complex hyperplane does not separate complex projective space into two components, because it has codimension 2. Category Complex manifolds Category Algebraic geometry Category Dimension mathanalysis stub ... more details
the analysis here. Polarity, tangent hyperplane, and singular points In general, a projective quadric ... P math to a hyperplane math p ast math of math P math , and vice versa, while preserving the incidence relation between points and hyperplanes. The coefficient vector of the polar hyperplane math h p ..., the hyperplane math h p ast math is well defined that is, not identically zero and does not contain math p math . If math p math is on the quadric and the hyperplane math h p ast math is well defined, and contains math p math which is said to be a regular point . In fact, it is the hyperplane ... hyperplane turns out to be the union of all lines that are either entirely contained in math Q math , or intersect math Q math at only one point. The condition for a point math u math to be in the hyperplane ... more details
Planes of existence The Logoic plane hyperplane or spacetime continuum universe , enclosing and interpenetrating grosser planes, respectively is the plane in which Brahman & Om Om or Aum , i.e. Logos or Agathon , i.e. according to Theosophy spirit of deity as Brahman, Logos i.e. Agathon, etc., i.e. the creative Word as the Pranava Om or Aum, Logos, Tikkun , etc. and ideal, exists. The term is from the Greek word Logos and is used in the Arcane School ideas. Theosophical differences Classical 1800s Theosophy does not say whether this spiritual plane is the only one that has to do with Logos is human, but Annie Besant & Charles Leadbeater may have said so, and Alice Bailey or others, who uses similar ideas, did say so. See also Adi metaphysical plane . References reflist DEFAULTSORT Divine Plane Category Esoteric cosmology Category Paranormal worlds and bodies ... more details
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