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Support 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 set, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reuleaux Supporting Lines
Reuleaux may refer to: * Franz Reuleaux (1829–1905), German mechanical engineer and lecturer * in geometry: ** Reuleaux polygon, a curve of constant width *** Reuleaux triangle, a Reuleaux polygon with three sides *** Reuleaux heptagon, a Reuleaux polygon with seven sides that provides the shape of some currency coins ** Reuleaux tetrahedron, the intersection of four spheres of equal radius centered at the vertices of a regular tetrahedron {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
00-02 Tau - Square - Reuleaux Wikipedia
The symbol , known in Unicode as hyphen-minus, is the form of hyphen most commonly used in digital documents. On most keyboards, it is the only character that resembles a minus sign or a dash, so it is also used for these. The name ''hyphen-minus'' derives from the original ASCII standard, where it was called ''hyphen (minus)''. The character is referred to as a ''hyphen'', a ''minus sign'', or a ''dash'' according to the context where it is being used. Description In early typewriters and character encodings, a single key/code was almost always used for hyphen, minus, various dashes, and strikethrough, since they all have a similar appearance. The current Unicode Standard specifies distinct characters for several different dashes, an unambiguous minus sign (sometimes called the ''Unicode minus'') at code point U+2212, an unambiguous hyphen (sometimes called the ''Unicode hyphen'') at U+2010, the hyphen-minus at U+002D and a variety of other hyphen symbols for various uses. W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...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, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's ''Elements'': "The urvedline 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 of an 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 article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), 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 Closure (mathematics), closed under the limit of a sequence, limit operation. This should not be confused with closed manifold. Sets that are both open and closed and are called clopen sets. Definition Given a topological space (X, \tau), the following statements are equivalent: # a set A \subseteq X is in X. # A^c = X \setminus A is an open subset of (X, \tau); that is, A^ \in \tau. # A is equal to its Closure (topology), closure in X. # A contains all of its limit points. # A contains all of its Boundary (topology), boundary points. An alternative characterization (mathematics), characterization of closed sets is available via sequences and Net (mathematics), net ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants. Affine geometry The affine transformations of the upper half-plane include # shifts (x,y)\mapsto (x+c,y), c\in\mathbb, and # dilations (x,y)\mapsto (\lambda x,\lambda y), \lambda > 0. Proposition: Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes A to B. :Proof: First shift the center of to Then take \lambda=(\text\ B)/(\text\ A) and dilate. Then shift to the center of Inversive geometry Definition: \mathcal := \left\ . can be recognized as the circle of radius centered at and as the polar plot of \rho(\theta) = \cos \theta. Proposition: in and are collinear points. In ... [...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, intuitively, 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 tangent to the 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. The point where the tangent line and the curve meet or intersect is called the ''point of tangency''. The tangent line is said to be "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 ... [...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 * '' Deutscher Michel'', a national personification of the German people 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 und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elena Deza
Elena Ivanovna Deza (, 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 Volgograd,. formerly Tsaritsyn. (1589–1925) and Stalingrad. (1925–1961), is the largest city and the administrative centre of Volgograd Oblast, Russia. The city lies on the western bank of the Volga, covering an area of , with a population ..., 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. 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 full professor in 2006. Books As well as m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supporting Hyperplane
In geometry, a supporting hyperplane of a Set (mathematics), 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 set, closed Half-space (geometry), 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 (topology), 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 (topology), interior such that every point on the boundary has a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull, 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 set, 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 projective duality, dual problem of intersecting Half-space (geome ... [...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 calculati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
