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.
If the defining polynomial of a plane algebraic curve is irreducible, then one has an irreducible plane algebraic curve. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its components, that are defined by the irreducible factors.
More generally, an algebraic curve is an algebraic variety of dimension one. In some contexts, an algebraic set of dimension one is also called an algebraic curve, but this will not be the case in this article. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an irreducible algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a birational equivalence.
These birational equivalences reduce most of the study of algebraic curves to the study of algebraic plane curves. However, some properties are not kept under birational equivalence and must be studied on non-plane curves. This is, in particular, the case for the degree and smoothness. For example, there exist smooth curves of genus 0 and degree greater than two, but any plane projection of such curves has singular points (see Genus–degree formula).
A non-plane curve is often called a space curve or a skew curve.
In Euclidean geometry An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation . 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 as a function of .
With a curve given by such an implicit equation, the first problems are to determine the shape of the curve and to draw it. These problems are not as easy to solve as in the case of the graph of a function, for which may easily be computed for various values of . The fact that the defining equation is a polynomial implies that the curve has some structural properties that may help in solving these problems.
Every algebraic curve may be uniquely decomposed into a finite number of smooth monotone arcs, called es, sometimes connected by some points sometimes called "remarkable points", and possibly a finite number of isolated points called acnodes. A smooth monotone arc is the graph of a smooth function which is defined and monotone on an open interval of the -axis. In each direction, an arc is either unbounded (usually called an infinite arc) or has an endpoint which is either a singular point (this will be defined below) or a point with a tangent parallel to one of the coordinate axes.
For example, for the Tschirnhausen cubic, there are two infinite arcs having the origin as of endpoint. This point is the only singular point of the curve. There are also two arcs having this singular point as one endpoint and having a second endpoint with a horizontal tangent. Finally, there are two other arcs each having one of these points with horizontal tangent as the first endpoint and having the unique point with vertical tangent as the second endpoint. In contrast, the sinusoid is certainly not an algebraic curve, having an infinite number of monotone arcs.
To draw an algebraic curve, it is important to know the remarkable points and their tangents, the infinite branches and their asymptotes (if any) and the way in which the arcs connect them. It is also useful to consider the inflection points as remarkable points. When all this information is drawn on a sheet of paper, the shape of the curve usually appears rather clearly. If not, it suffices to add a few other points and their tangents to get a good description of the curve.
The methods for computing the remarkable points and their tangents are described below in the section Remarkable points of a plane curve.
Plane projective curves
It is often desirable to consider curves in the projective space. An algebraic curve in the projective plane or plane projective curve is the set of the points in a projective plane whose projective coordinates are zeros of a homogeneous polynomial in three variables .
Every affine algebraic curve of equation may be completed into the projective curve of equation , where
is the result of the homogenization of . Conversely, if is the homogeneous equation of a projective curve, then is the equation of an affine curve, which consists of the points of the projective curve whose third projective coordinate is not zero. These two operations are reciprocal one to the other, as
and, if is defined by , then
as soon as the homogeneous polynomial is not divisible by .
For example, the projective curve of equation
is the projective completion of the unit circle of equation
This implies that an affine curve and its projective completion are the same curves, or, more precisely that the affine curve is a part of the projective curve that is large enough to well define the "complete" curve. This point of view is commonly expressed by calling "points at infinity" of the affine curve the points (in finite number) of the projective completion that do not belong to the affine part.
Projective curves are frequently studied for themselves. They are also useful for the study of affine curves. For example, if is the polynomial defining an affine curve, beside the partial derivatives and , it is useful to consider the derivative at infinity
For example, the equation of the tangent of the affine curve of equation at a point is
Remarkable points of a plane curve
In this section, we consider a plane algebraic curve defined by a bivariate polynomial p(x, y) and its projective completion, defined by the homogenization of p.
Intersection with a line Knowing the points of intersection of a curve with a given line is frequently useful. The intersection with the axes of coordinates and the asymptotes are useful to draw the curve. Intersecting with a line parallel to the axes allows one to find at least a point in each branch of the curve. If an efficient root-finding algorithm is available, this allows to draw the curve by plotting the intersection point with all the lines parallel to the y-axis and passing through each pixel on the x-axis.
If the polynomial defining the curve has a degree d, any line cuts the curve in at most d points. Bézout's theorem asserts that this number is exactly d, if the points are searched in the projective plane over an algebraically closed field (for example the complex numbers), and counted with their multiplicity. The method of computation that follows proves again this theorem, in this simple case.
To compute the intersection of the curve defined by the polynomial p with the line of equation ax+by+c = 0, one solves the equation of the line for x (or for y if a = 0). Substituting the result in p, one gets a univariate equation q(y) = 0 (or q(x) = 0, if the equation of the line has been solved in y), each of whose roots is one coordinate of an intersection point. The other coordinate is deduced from the equation of the line. The multiplicity of an intersection point is the multiplicity of the corresponding root. There is an intersection point at infinity if the degree of q is lower than the degree of p; the multiplicity of such an intersection point at infinity is the difference of the degrees of p and q.
Tangent at a point The tangent at a point (a, b) of the curve is the line of equation , like for every differentiable curve defined by an implicit equation. In the case of polynomials, another formula for the tangent has a simpler constant term and is more symmetric:
where is the derivative at infinity. The equivalence of the two equations results from Euler's homogeneous function theorem applied to P.
If the tangent is not defined and the point is a singular point.
This extends immediately to the projective case: The equation of the tangent of at the point of projective coordinates (a:b:c) of the projective curve of equation P(x, y, z) = 0 is
and the points of the curves that are singular are the points such that
(The condition P(a, b, c) = 0 is implied by these conditions, by Euler's homogeneous function theorem.)
Asymptotes Every infinite branch of an algebraic curve corresponds to a point at infinity on the curve, that is a point of the projective completion of the curve that does not belong to its affine part. The corresponding asymptote is the tangent of the curve at that point. The general formula for a tangent to a projective curve may apply, but it is worth to make it explicit in this case.
Let be the decomposition of the polynomial defining the curve into its homogeneous parts, where pi is the sum of the monomials of p of degree i. It follows that
and
A point at infinity of the curve is a zero of p of the form (a, b, 0). Equivalently, (a, b) is a zero of pd. The fundamental theorem of algebra implies that, over an algebraically closed field (typically, the field of complex numbers), pd factors into a product of linear factors. Each factor defines a point at infinity on the curve: if bx − ay is such a factor, then it defines the point at infinity (a, b, 0). Over the reals, pd factors into linear and quadratic factors. The irreducible quadratic factors define non-real points at infinity, and the real points are given by the linear factors. If (a, b, 0) is a point at infinity of the curve, one says that (a, b) is an asymptotic direction. Setting q = pd the equation of the corresponding asymptote is
If and the asymptote is the line at infinity, and, in the real case, the curve has a branch that looks like a parabola. In this case one says that the curve has a parabolic branch. If
the curve has a singular point at infinity and may have several asymptotes. They may be computed by the method of computing the tangent cone of a singular point.
Singular points The singular points of a curve of degree d defined by a polynomial p(x,y) of degree d are the solutions of the system of equations:
In characteristic zero, this system is equivalent to
where, with the notation of the preceding section, The systems are equivalent because of Euler's homogeneous function theorem. The latter system has the advantage of having its third polynomial of degree d-1 instead of d.
Similarly, for a projective curve defined by a homogeneous polynomial P(x,y,z) of degree d, the singular points have the solutions of the system
as homogeneous coordinates. (In positive characteristic, the equation has to be added to the system.)
This implies that the number of singular points is finite as long as p(x,y) or P(x,y,z) is square free. Bézout's theorem implies thus that the number of singular points is at most (d − 1)2, but this bound is not sharp because the system of equations is overdetermined. If reducible polynomials are allowed, the sharp bound is d(d − 1)/2, this value is reached when the polynomial factors in linear factors, that is if the curve is the union of d lines. For irreducible curves and polynomials, the number of singular points is at most (d − 1)(d − 2)/2, because of the formula expressing the genus in term of the singularities (see below). The maximum is reached by the curves of genus zero whose all singularities have multiplicity two and distinct tangents (see below).