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Algebraic Curve
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 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 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. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a ... [...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, von 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 R C 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 form for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus–degree Formula
In classical algebraic geometry, the genus–degree formula relates the degree ''d'' of an irreducible plane curve C with its arithmetic genus ''g'' via the formula: :g=\frac12 (d-1)(d-2). Here "plane curve" means that C is a closed curve in the projective plane \mathbb^2. If the curve is non-singular the geometric genus and the arithmetic genus are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinary singularity of multiplicity ''r'' decreases the genus by \frac12 r(r-1). Proof The proof follows immediately from the adjunction formula. For a classical proof see the book of Arbarello, Cornalba, Griffiths and Harris. Generalization For a non-singular hypersurface H of degree ''d'' in the projective space \mathbb^n of arithmetic genus ''g'' the formula becomes: : g=\binom , \, where \tbinom is the binomial coefficient. Notes See also * Thom conjecture References * * Enrico Arbarello, Maur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to Vertical asymptotes are vertical lines near which the fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine Wave
A sine wave, sinusoidal wave, or just sinusoid is a curve, mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph of a function, graph. It is a type of continuous wave and also a Smoothness, smooth periodic function. It occurs often in mathematics, as well as in physics, engineering, signal processing and many other fields. Formulation Its most basic form as a function of time (''t'') is: y(t) = A\sin(2 \pi f t + \varphi) = A\sin(\omega t + \varphi) where: * ''A'', ''amplitude'', the peak deviation of the function from zero. * ''f'', ''frequency, ordinary frequency'', the ''Real number, number'' of oscillations (cycles) that occur each second of time. * ''ω'' = 2''f'', ''angular frequency'', the rate of change of the function argument in units of radians per second. * \varphi, ''phase (waves), phase'', specifies (in radians) where in its cycle the oscillation is at ''t'' = 0. When \varphi is non-zero, the entire waveform appears to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
