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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.
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.
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 ...
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 ).
Let A : (a,b) → R 2 be a parametric plane curve, in coordinates A(t) = (x(t),y(t)), and B be another (unparameterized) curve. Suppose, as before, that the curve A tends to infinity. The curve B is a curvilinear asymptote of A if the shortest distance from the point A(t) to a point on B tends to zero as t → b.
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.
An algebraic variety that has no singular point is said to be non-singular or smooth. The concept is generalized to smooth schemes in the modern language of scheme theory. The plane algebraic curve (a cubic curve) of equation y 2 − x 2 (x + 1) = 0 crosses itself at the origin (0, 0). The origin is a double point of this curve.