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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.
An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation (,) = (or (,,) =, where F is a homogeneous polynomial, in the projective case.) Algebraic curves have been studied extensively since the 18th century.
The affine plane curve y 2 = x 3 − x. The corresponding projective curve is called an elliptic curve. A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. The projective line P 1 is an example of a projective curve; it can be viewed as the curve in the projective plane P 2 = {[x, y, z ...
Algebraic curves in the plane may be defined as the set of points (x, y) satisfying an equation of the form (,) =, where f is a polynomial function :. If f is expanded as = + + + + + + If the origin (0, 0) is on the curve then a 0 = 0.
A space curve is a curve for which is at least three-dimensional; a skew curve is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to real algebraic curves , although the above definition of a curve does not apply (a real algebraic curve may be disconnected ).
For example, the definition of elliptic curve from algebraic geometry is connected non-singular projective curve of genus 1 with a given rational point on it. By the Riemann–Roch theorem , an irreducible plane curve of degree d {\displaystyle d} given by the vanishing locus of a section s ∈ Γ ( P 2 , O P 2 ( d ) ) {\displaystyle s\in ...
In classical algebraic geometry, the genus–degree formula relates the degree of an irreducible plane curve with its arithmetic genus via the formula: = (). Here "plane curve" means that is a closed curve in the projective plane.
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation F ( x , y , z ) = 0 {\displaystyle F(x,y,z)=0} applied to homogeneous coordinates ( x : y : z ) {\displaystyle (x:y:z)} for the projective plane ; or the inhomogeneous version for the affine space determined by setting z = 1 in such an ...