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Algebraic Curves
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenization of a polynomial, homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse function, inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. If the defining polynomial of a plane algebraic curve is irreducible polynomial, irreducible, then one has an ''irreducible plane algebraic curve''. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its ''Irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tschirnhausen Cubic
In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation :r = a\sec^3 \left(\frac\right) where is the secant function. History The curve was studied by Ehrenfried Walther von Tschirnhaus, Tschirnhaus, Guillaume de l'Hôpital, de L'Hôpital, and Eugène Charles Catalan, Catalan. It was given the name Tschirnhausen cubic in a 1900 paper by Raymond Clare Archibald, though it is sometimes known as de L'Hôpital's cubic or the trisectrix of Catalan. Other equations Put t=\tan(\theta/3). Then applying De Moivre's formula, triple-angle formulas gives :x=a\cos \theta \sec^3 \frac = a \left(\cos^3 \frac - 3 \cos \frac \sin^2 \frac \right) \sec^3 \frac= a\left(1 - 3 \tan^2 \frac\right) ::= a(1 - 3t^2) :y=a\sin \theta \sec^3 \frac = a \left(3 \cos^2 \frac\sin \frac - \sin^3 \frac \right) \sec^3 \frac= a \left(3 \tan \frac - \tan^3 \frac \right) ::= at(3-t^2) giving a parametric equation, parametric for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
