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Ampersand Curve
In geometry, the ampersand curve is a type of quartic plane curve. It was named after its resemblance to the &, ampersand symbol by Martyn Cundy, Henry Cundy and Arthur Rollett. The ampersand curve is the graph of the equation :6x^4+4y^4-21x^3+6x^2y^2+19x^2-11xy^2-3y^2=0. The graph of the ampersand curve has three crunode points where it intersects itself at (0,0), (1,1), and (1,-1). The curve has a Genus (mathematics), genus of 0. The curve was originally constructed by Julius Plücker as a quartic plane curve that has 28 real bitangents, the maximum possible for bitangents of a quartic. It is the special case of the Plücker quartic :(x+y)(y-x)(x-1)(x-\tfrac)-2(y^2+x(x-2))^2-k=0, with k=0. The curve has 6 real horizontal tangents at *\left(\frac, \pm\frac\right), *\left(\frac, \pm\frac\right), and *\left(\frac, \pm\frac\right). And 4 real vertical tangents at \left(-\tfrac,\pm\tfrac\right) and \left(\tfrac,\tfrac\right). It is an example of a curve that has no value of x in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quartic Plane Curve
In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one of not equal to zero. This equation has 15 constants. However, it can be multiplied by any non-zero constant without changing the curve; thus by the choice of an appropriate constant of multiplication, any one of the coefficients can be set to 1, leaving only 14 constants. Therefore, the space of quartic curves can be identified with the real projective space It also follows, from Cramer's theorem on algebraic curves, that there is exactly one quartic curve that passes through a set of 14 distinct points in general position, since a quartic has 14 degrees of freedom. A quartic curve can have a maximum of: * Four connected components * Twenty-eight bi-tangents * Three ordinary double points. One may also consider quartic curves over oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martyn Cundy
Henry Martyn Cundy (23 December 1913 – 25 February 2005) was a mathematics teacher and professor in Britain and Malawi as well as a singer, musician and poet. He was one of the founders of the School Mathematics Project to reform O level and A level teaching. Through this he had a big effect on maths teaching in Britain and especially in Africa. Education and career Cundy attended Monkton Combe School and then read mathematics at Trinity College, Cambridge, where he earned a PhD in quantum theory in 1938. In 1937, Cundy was awarded the Cambridge University Rayleigh Prize for Mathematical Physics (now known as the Rayleigh-Knight Prize) for an essay entitled "Motion in a Tetrahedral Field". Others awarded the Rayleigh Prize include Royal Society Fellows Alan Turing and Fred Hoyle; instead of acquiring a University position, Cundy initially chose work at the secondary school level. He taught at the Sherborne School from 1938 to 1966 and became prominently involved in the refo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crunode
In mathematics, a crunode (archaic; from Latin ''crux'' "cross" + ''node'') or node of an algebraic curve is a type of singular point at which the curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ''ordinary double point''. In the case of a smooth real plane curve , a point is a crunode provided that both first partial derivatives vanish \frac = \frac = 0 and the Hessian determinant is negative: \frac \frac - \left(\frac\right)^2 < 0. See also * * Acnode * * ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus (mathematics)
In mathematics, genus (: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic \chi, via the relationship \chi=2-2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads \chi=2-2g-b. In layman's terms, the genus is the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julius Plücker
Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the discovery of the electron. He also vastly extended the study of Lamé curves. Biography Early years Plücker was born at Elberfeld (now part of Wuppertal). After being educated at Düsseldorf and at the universities of Bonn, Heidelberg and Berlin he went to Paris in 1823, where he came under the influence of the great school of French geometers, whose founder, Gaspard Monge, had only recently died. In 1825 he returned to Bonn, and in 1828 was made professor of mathematics. In the same year he published the first volume of his ''Analytisch-geometrische Entwicklungen'', which introduced the method of "abridged notation". In 1831 he published the second volume, in which he clearly established on a firm and independent basis projecti ... [...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] |
Bitangents Of A Quartic
In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for which all 28 of these lines have real numbers as their coordinates and therefore belong to the Euclidean plane. An explicit quartic with twenty-eight real bitangents was first given by As Plücker showed, the number of real bitangents of any quartic must be 28, 16, or a number less than 9. Another quartic with 28 real bitangents can be formed by the locus of centers of ellipses with fixed axis lengths, tangent to two non-parallel lines. gave a different construction of a quartic with twenty-eight bitangents, formed by projecting a cubic surface; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is the line at infinity in the projective plane. Example The Trott curve, another curve with 28 real bitangents, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain (math)
A domain is a geographic area controlled by a single person or organization. Domain may also refer to: Law and human geography * Demesne, in English common law and other Medieval European contexts, lands directly managed by their holder rather than being delegated to subordinate managers * Domaine, a large parcel of land under single ownership, which would historically generate income for its owner. * Eminent domain, the right of a government to appropriate another person's property for public use * Private domain / Public domain, places defined under Jewish law where it is either permitted or forbidden to move objects on the Sabbath day * Public domain, creative work to which no exclusive intellectual property rights apply * Territory (subdivision), a non-sovereign geographic area which has come under the authority of another government Science * Domain (biology), a taxonomic subdivision larger than a kingdom * Domain of discourse, the collection of entities of interest in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
