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Arc (projective Geometry)
A (simple) arc in finite projective geometry is a set of points which satisfies, in an intuitive way, a feature of ''curved'' figures in continuous geometry, continuous geometries. Loosely speaking, they are sets of points that are far from "line-like" in a plane or far from "plane-like" in a three-dimensional space. In this finite setting it is typical to include the number of points in the set in the name, so these simple arcs are called -arcs. An important generalization of -arcs, also referred to as arcs in the literature, is the ()-arcs. -arcs in a projective plane In a finite projective plane (not necessarily Desarguesian) a set of points such that no three points of are Collinear points, collinear (on a line) is called a . If the plane has order then , however the maximum value of can only be achieved if is even. In a plane of order , a -arc is called an Oval (projective plane), oval and, if is even, a -arc is called a Oval (projective plane), hyperoval. Every con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperoval In Fano Plane
In projective geometry an oval is a point set in a plane that is defined by Incidence (geometry), incidence properties. The standard examples are the nondegenerate Conic section, conics. However, a conic is only defined in a pappian plane, whereas an oval may exist in any type of projective plane. In the literature, there are many criteria which imply that an oval is a conic, but there are many examples, both infinite and finite, of ovals in pappian planes which are not conics. As mentioned, in projective geometry an oval is defined by incidence properties, but in other areas, ovals may be defined to satisfy other criteria, for instance, in differential geometry by differentiability conditions in the real plane. The higher dimensional analog of an oval is an Ovoid (projective geometry), ovoid in a projective space. A generalization of the oval concept is an abstract oval, which is a structure that is not necessarily embedded in a projective plane. Indeed, there exist abstract o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Geometry
A finite geometry is any geometry, geometric system that has only a finite set, finite number of point (geometry), points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective space, projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius plane, Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometry, inversive geometries. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Rational Curve
In mathematics, the rational normal curve is a smooth, rational curve of degree in projective n-space . It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For it is the plane conic and for it is the twisted cubic. The term "normal" refers to projective normality, not normal schemes. The intersection of the rational normal curve with an affine space is called the moment curve. Definition The rational normal curve may be given parametrically as the image of the map :\nu:\mathbf^1\to\mathbf^n which assigns to the homogeneous coordinates the value :\nu: :T\mapsto \left ^n:S^T:S^T^2:\cdots:T^n \right In the affine coordinates of the chart the map is simply :\nu:x \mapsto \left (x, x^2, \ldots, x^n \right ). That is, the rational normal curve is the closure by a single point at infinity of the affine curve :\left (x, x^2, \ldots, x^n \right ). Equivalently, rational normal curve may be under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Arc
A maximal arc in a finite projective plane is a largest possible (''k'',''d'')- arc in that projective plane. If the finite projective plane has order ''q'' (there are ''q''+1 points on any line), then for a maximal arc, ''k'', the number of points of the arc, is the maximum possible (= ''qd'' + ''d'' − ''q'') with the property that no ''d''+1 points of the arc lie on the same line. Definition Let \pi be a finite projective plane of order ''q'' (not necessarily desarguesian). Maximal arcs of ''degree'' ''d'' (2 ≤ ''d'' ≤ ''q'' − 1) are (''k'',''d'')- arcs in \pi, where ''k'' is maximal with respect to the parameter ''d'', in other words, ''k'' = ''qd'' + ''d'' − ''q''. Equivalently, one can define maximal arcs of degree ''d'' in \pi as non-empty sets of points ''K'' such that every line intersects the set either in 0 or ''d'' points. Some authors permit the degree of a maximal arc to be 1, ''q'' or even ''q'' + 1. Letting ''K'' be a maximal (''k'', ''d'')-arc in a proj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane (geometry)
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line. Most commonly, the ambient space is -dimensional Euclidean space, in which case the hyperplanes are the -dimensional "flats", each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion (geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection (set theory), intersection of half-spaces. In non-Euclidean geometry, the ambient space might be the n-sphere, -dimensional sphere or hyperbolic space, or more generally a pseudo-Riem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the 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 be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Segre's Theorem
In projective geometry, Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement: *Any Oval (projective plane), oval in a ''finite Pappus's hexagon theorem, pappian'' projective plane of ''odd'' order is a nondegenerate projective conic section. This statement was assumed 1949 by the two Finnish mathematicians Gustaf Järnefelt, G. Järnefelt and Paul Kustaanheimo, P. Kustaanheimo and its proof was published in 1955 by B. Segre. A finite pappian projective plane can be imagined as the projective closure of the real plane (by a line at infinity), where the real numbers are replaced by a finite field . ''Odd order'' means that is odd. An oval is a curve similar to a circle (see definition below): any line meets it in at most 2 points and through any point of it there is exactly one tangent. The standard examples are the nondegenerate projective conic sections. In pappian projective planes of ''even'' order greater than four there are ovals which are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''projective space'') and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "Point at infinity, points at infinity") to Euclidean points, and vice versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translation (geometry), translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in Euclidean geometry, the concept of an angle does not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beniamino Segre
Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of finite geometry. Life and career He was born and studied in Turin. Corrado Segre, his uncle, also served as his doctoral advisor. Among his main contributions to algebraic geometry are studies of birational invariants of algebraic varieties, singularities and algebraic surfaces. His work was in the style of the old Italian School, although he also appreciated the greater rigour of modern algebraic geometry. Segre was a pioneer in finite geometry, in particular projective geometry based on vector spaces over a finite field. In a well-known paper he proved the following theorem: In a Desarguesian plane of odd order, the ovals are exactly the irreducible conics. In 1959 he authored a survey "Le geometrie di Galois" on Galois geometry. According to J. W. P. Hirschfeld, it "gave a comprehensive list of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oval (projective 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. However, a conic is only defined in a pappian plane, whereas an oval may exist in any type of projective plane. In the literature, there are many criteria which imply that an oval is a conic, but there are many examples, both infinite and finite, of ovals in pappian planes which are not conics. As mentioned, in projective geometry an oval is defined by incidence properties, but in other areas, ovals may be defined to satisfy other criteria, for instance, in differential geometry by differentiability conditions in the real plane. The higher dimensional analog of an oval is an ovoid in a projective space. A generalization of the oval concept is an abstract oval, which is a structure that is not necessarily embedded in a projective plane. Indeed, there exist abstract ovals which can not lie in any projective plane. Definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
