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An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
Fermat curve; Bézout's theorem; Brill–Noether theory; Genus (mathematics) Riemann surface; Riemann–Hurwitz formula; Riemann–Roch theorem; Abelian integral; Differential of the first kind; Jacobian variety. Generalized Jacobian; Moduli of algebraic curves; Hurwitz's theorem on automorphisms of a curve; Clifford's theorem on special divisors
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials ; the modern approach generalizes this in a few different aspects.
It relies on the axiomatic method for proving all results from a few basic properties initially called postulates, and at present called axioms. After the 17th-century introduction by René Descartes of the coordinate method, which was called analytic geometry , the term "synthetic geometry" was coined to refer to the older methods that were ...
In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class.In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore L such that L 2 is the canonical bundle, here also equivalently the holomorphic cotangent bundle.
Rational Bézier curve – polynomial curve defined in homogeneous coordinates (blue) and its projection on plane – rational curve (red) In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, [1] [2] [3] are a system of coordinates used in projective geometry, just as Cartesian coordinates are used ...
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