Cubic plane curve


In mathematics, a cubic plane curve , often called simply a cubic is a plane algebraic curve defined by a homogeneous polynomial of degree 3 in three variables or by the corresponding polynomial in two variables Starting from, one can recover as.
Typically, the coefficients of the polynomial belong to but they may belong to any field, in which case, one talks of a cubic defined over. The points of the cubic are the points of the projective space of dimension three over the field of the complex numbers, whose projective coordinates satisfy the equation of the cubic
A point at infinity of the cubic is a point such that. A real point of the cubic is a point with real coordinates. A point defined over is a point with coordinates in.
Generally, the defining polynomial is implicitly assumed to be irreducible, since, otherwise, the equation defines either three lines, or a conic section and a line. However, it is often convenient to include the decomposed curves into the cubics. When the distinction is needed, one talks of irreducible cubics and decomposed cubics.

Basics

A cubic plane curve, or simply a cubic is basically the set of the points in the Euclidean plane whose Cartesian coordinates are zeros of a polynomial of degree 3 in two variables:
Typically, the coefficients are real numbers, and the points of the cubic are real zeros of. The nonreal complex zeros of are also considered as points of the cubic, and the points in the Euclidean plane are called real points of the cubic to distinguish them from the nonreal ones.
It is common and often needed for technical reasons to extend the cubic defined by to the projective plane, by considering as points of the cubic the points of the projective plane whose projective coordinates satisfy, where
The points of the Euclidean plane are identified with the points of the projective plane with by the relation. The points of the cubic such that are called the points at infinity of the cubic.
Everything that precedes applies by replacing the field of the real numbers with any field, the Euclidean plane with an affine plane over,, the complex numbers with an algebraically closed field containing, "real point" with "point defined over " or "-point", etc.
A cubic is degenerated or decomposed if the polynomial is not absolutely irreducible. In this case, either there is an irreducible factor of degree 2 and the cubic is decomposed into a conic and a line, or there are three linear factors corresponding to the decomposition of the cubic into three lines that are not necessarily distinct. A non-degenerated cubic is called an irreducible cubic.
In the projective plane over the algebraically closed field, every line intersects the conic in three points, not necessarily distinct.

Tangents and singular points

The equation of the tangent at a point of projective coordinates on the cubic is
If all three partial derivatives at are equal to zero, the tangent is undefined, and the point is a singular point.
An irreducible cubic has at most one singular point, since otherwise the line passing through two singular points would intersect the cubic at four points.
The singular points of a decomposed cubic are the intersection points of two components, and, if any, all points of a multiple component.
If an irreducible cubic has a singular point of projective coordinates,the tangent cone consists of two lines that are distinct of not. If the tangent cone is a double line, the singular point is a cusp. Otherwise, it is an ordinary double point.
Over the reals, such an ordinary point may be either a crunode if the two tangent lines are real, or an acnode if they are complex conjugate. When the real points of the curve are plotted, an acnode appear as an isolated point, a crunode appears as a point where the curve crosses itself, and a cusp appears a point where a moving point must reverse direction.

Inflection points

An inflection point is a regular point of a curve where the tangent has a contact of order at least 3, and thus exactly 3 in the case of cubic curves. The inflection points of an algebraic plane curve are common zeros of the projective equation of the curve
and the Hessian determinant
In the case of a cubic, both polynomials are of degree 3, and by Bézout's theorem, there are at most 9 inflexion points over an algebraic closure of the field of definition of the cubic. More precisely, the common zeros are the inflection points are the common zeros. The inflection points are the common zeros of multiplicity one, and the singular points, if any, are the common zeros of higher multiplicity.
A cubic with a cusp has exactly one inflection point. A cubic with an ordinary double point has three colinear inflection points; over the reals, the three inflection points are real if the singular point is an acnode; if it is a crunode, there is a real inflection point and two complex conjugate ones. A non-singular cubic has 9 inflection points that have a special configuration ; over the reals, exactly 3 of the inflection points are real, and they are colinear.

Real shapes

Real cubics may have many shapes in a Euclidean plane.
In Weierstrass normal form
their shape depends from the parameters and, and, more specifically on the signs of,, and :
  • If, the cubic is singular.
  • *If, the singular point is a cusp.
  • *If and there is a acnode.
  • *If and there is a crunode.
  • If, the cubic is non-singular and has a single unbounded branch; the sign of determines whether there are tangents parallel to the -axis.
  • If, the cubic is non-singular, and has an unbounded branch and an "oval".
In the following plots, the singular point is placed at the origin. Except for the semicubical parabola, a translation of 1/3 to the left or to the right is needed for having a true Weierstrass form.
For the cubics that are not in Weierstrass normal form, the shape depends on the shape of the corresponding Weierstrass form and on the configuration of the intersection with the line at infinity.

Singular cubics

An irreducible cubic is said to be singular if it has a singular point in the projective plane, even if it has none in the Euclidean plane.
In particular, the graph of a cubic function is regular in the Euclidean plane but has a singular point at infinity in the direction of the -axis. This point is a cusp with the line at infinity as its double tangent. Other examples of singular cubics that are regular in the Euclidean plane are the trident curve with a double point at infinity and the witch of Agnesi with an isolated point at infinity. All these cubics are special cases of the singular cubics of equation, where and are polynomials in such that.
Examples of cubics that have a double point in the Euclidean plane are the folium of Descartes, the Tschirnhausen cubic, and the trisectrix of Maclaurin. Example with a cusp are the semicubical parabola and the cissoid of Diocles. The curve is an example having an isolated point at the origin.
Singular cubics are also called unicursal cubics, because a moving point travelling the cubic can cover the whole cubic in a single course. They are the rational cubics, that is the cubics that admit a rational parametrization, a parametrization in terms of rational functions.
Indeed, the lines passing through the singular point depend on a single parameter, which can be the slope in the Euclidean plane. The three intersection points of the cubic and such a line consist of twice the singular point and a single other point whose coordinate can be obtained by solving a linear equation.
More precisely, given a singular conic, one may change coordinates for having the singular point at the origin. Then the equation of the cubic has the form
where and are homogeneous polynomials of respective degrees 3 and 2. Setting, one gets
giving the parametric equation
If desired, one can make back the change of coordinates for having the parametrization in terms of the original coordinates.
Conversely, if,, and are three polynomials without a common factor, that have 3 as their maximal degree, then the parametric equation
defines a singular cubic whose implicit equation can be obtained as the resultant

Weierstrass normal form

Over a field of characteristic different from 2 and 3, every irreducible cubic can be transformed into the Weierstrass normal form
by a projective transformation, or equivalently by a change of projective coordinates. The parameters and may belong to the field of definition of the cubic even if the projective transformation may require to work over an algebraic extension of the field of definition. Over the real numbers, a real projective transformation is always possible.
For this change of coordinates one can proceed as follows.
Firstly, choose an inflection point and a projective coordinate system such that the inflection point is at infinity in the direction of the -axis, with the line at infinity as its tangent. Over the real, there is always a real inflexion point, and the projective transformation is real. Over other fields, it may be that an algebraic field extension is needed. The resulting equation has the form
One has, since, otherwise, the line at infinity would be a component of the curve. One has also, since otherwise, the point would be a singular point and thus not an inflexion point.
The transformation and the division of the whole equation by allows supposing. The transformation gives . Finally, the transformation gives the Weierstrass normal form.
The Weierstrass normal form is not unique since the transformation and the multiplication of the whole equation by amounts to multiply the coefficient of and the constant coefficient by and respectively.
The invariant theory shows that no other Weierstrass normal forms exist for a given cubic, even if one changes the initial choice of an inflection point. Moreover, even if the inflexion point is not defined over the field of definition of the cubic, one can choose for getting a Weierstrass normal form with coefficients in the field of definition of the cubic.