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Supporting Line
In geometry, a supporting line ''L'' of a curve ''C'' in the plane is a line that contains a point of ''C'', but does not separate any two points of ''C''."The geometry of geodesics", Herbert Busemannp. 158/ref> In other words, ''C'' lies completely in one of the two closed half-planes defined by ''L'' and has at least one point on ''L''. Properties There can be many supporting lines for a curve at a given point. When a tangent exists at a given point, then it is the unique supporting line at this point, if it does not separate the curve. Generalizations The notion of supporting line is also discussed for planar shapes. In this case a supporting line may be defined as a line which has common points with the boundary of the shape, but not with its interior."Encyclopedia of Distances", by Michel M. Deza, Elena Dezap. 179/ref> The notion of a supporting line to a planar curve or convex shape can be generalized to n dimension as a supporting hyperplane. Critical support lines If tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geometry), point. This is the definition that appeared more than 2000 years ago in Euclid's Elements, Euclid's ''Elements'': "The [curved] line is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image (mathematics), image of an interval (mathematics), interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this artic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a '' tangent line approximation'', the graph of the affine function that best approximates the original function at the given point. Similarly, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michel M
Michel may refer to: * Michel (name), a given name or surname of French origin (and list of people with the name) * Míchel (nickname), a nickname (a list of people with the nickname, mainly Spanish footballers) * Míchel (footballer, born 1963), Spanish former footballer and manager * ''Michel'' (TV series), a Korean animated series * German auxiliary cruiser ''Michel'' * Michel catalog, a German-language stamp catalog * St. Michael's Church, Hamburg or Michel * S:t Michel, a Finnish town in Southern Savonia, Finland People * Alain Michel (other), several people * Ambroise Michel (born 1982), French actor, director and writer. * André Michel (director), French film director and screenwriter * André Michel (lawyer), human rights and anti-corruption lawyer and opposition leader in Haiti * Anette Michel (born 1971), Mexican actress * Anneliese Michel (1952 - 1976), German Catholic woman undergone exorcism * Annett Wagner-Michel (born 1955), German Woman International ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elena Deza
Elena Ivanovna Deza (russian: Елена Ивановна Деза, née Panteleeva; born 23 August 1961) is a French and Russian mathematician known for her books on metric spaces and figurate numbers. Education and career Deza was born on 23 August 1961 in Volgograd, and is a French and Russian citizen. She earned a diploma in mathematics in 1983, a candidate's degree (doctorate) in mathematics and physics in 1993, and a docent's certificate in number theory in 1995, all from Moscow State Pedagogical University Moscow State Pedagogical University or Moscow State University of Education is an educational and scientific institution in Moscow, Russia, with eighteen faculties and seven branches operational in other Russian cities. The institution had underg .... From 1983 to 1988, Deza was an assistant professor of mathematics at Moscow State Forest University. In 1988 she moved to Moscow State Pedagogical University; she became a lecturer there in 1993, a reader in 1994, and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supporting Hyperplane
In geometry, a supporting hyperplane of a set S in Euclidean space \mathbb R^n is a hyperplane that has both of the following two properties: * S is entirely contained in one of the two closed half-spaces bounded by the hyperplane, * S has at least one boundary-point on the hyperplane. Here, a closed half-space is the half-space that includes the points within the hyperplane. Supporting hyperplane theorem This theorem states that if S is a convex set in the topological vector space X=\mathbb^n, and x_0 is a point on the boundary of S, then there exists a supporting hyperplane containing x_0. If x^* \in X^* \backslash \ (X^* is the dual space of X, x^* is a nonzero linear functional) such that x^*\left(x_0\right) \geq x^*(x) for all x \in S, then :H = \ defines a supporting hyperplane. Conversely, if S is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then S is a convex set. The hyperplane in the theorem may not be uniqu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitangent
In geometry, a bitangent to a curve is a line that touches in two distinct points and and that has the same direction as at these points. That is, is a tangent line at and at . Bitangents of algebraic curves In general, an algebraic curve will have infinitely many secant lines, but only finitely many bitangents. Bézout's theorem implies that an algebraic plane curve with a bitangent must have degree at least 4. The case of the 28 bitangents of a quartic was a celebrated piece of geometry of the nineteenth century, a relationship being shown to the 27 lines on the cubic surface. Bitangents of polygons The four bitangents of two disjoint convex polygons may be found efficiently by an algorithm based on binary search in which one maintains a binary search pointer into the lists of edges of each polygon and moves one of the pointers left or right at each steps depending on where the tangent lines to the edges at the two pointers cross each other. This bitangent calculation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annulus (mathematics)
In mathematics, an annulus (plural annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word ''anulus'' or ''annulus'' meaning 'little ring'. The adjectival form is annular (as in annular eclipse). The open annulus is topologically equivalent to both the open cylinder and the punctured plane. Area The area of an annulus is the difference in the areas of the larger circle of radius and the smaller one of radius : :A = \pi R^2 - \pi r^2 = \pi\left(R^2 - r^2\right). The area of an annulus is determined by the length of the longest line segment within the annulus, which is the chord tangent to the inner circle, in the accompanying diagram. That can be shown using the Pythagorean theorem since this line is tangent to the smaller circle and perpendicular to its radius at that point, so and are sides of a right-angled triangle with hypotenuse , and the ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
