Cubic plane curve

"Cubic curve" redirects here. For information on polynomial functions of degree 3, see Cubic function.

In mathematics, a cubic plane curve is a plane algebraic curve C defined by a homogeneous cubic equation

⁠${\displaystyle F(x,y,z)=0}$⁠

A selection of cubic curves

applied to the three homogeneous coordinates of the projective plane, or the corresponding equation for Cartesian coordinates in the Euclidean plane determined by setting ⁠${\displaystyle z=1}$⁠ in such an equation.

Singular cubic y² = x² ⋅ (x + 1). A parametrization is given by t ↦ (t² – 1, t ⋅ (t² – 1)).

Classification by singularities

A cubic curve may have a singular point, in which case it has a parametrization in terms of a projective line. The singular points of an irreducible plane cubic curve are quite limited: one double point, one cusp, or one isolated point. Singular irreducible plane cubic curves include the folium of Descartes, the Tschirnhausen cubic, and the trisectrix of Maclaurin (all having a double point) and the semicubical parabola and cissoid of Diocles (both having a cusp). The curve ${\displaystyle y^{2}=x^{2}(x-1)}$ provides an example having an isolated point, at the origin.^[1] A cubic curve may be non-singular in the Euclidean plane, while having a singular point in the projective plane. This is the case for the cubic parabola (the graph of the cube function), which has a cusp at infinity, for the trident curve with a double point at infinity, and for the witch of Agnesi, which has an isolated point at infinity.

A reducible plane cubic curve is either a conic and a line or three lines. For a conic and a line, its singularities may consist either of two double points where the conic crosses the line, or a tacnode where the conic is tangent to the line. Three distinct lines may have up to three double points, or a single triple point (concurrent lines).

Every irreducible cubic curve can be transformed by a projective transformation into the special form ${\displaystyle y^{2}=x^{3}+ax^{2}+bx+c}$.^[2] A cubic curve in this form is singular if and only if the cubic polynomial in ⁠${\displaystyle x}$⁠ has a double or a triple root. A nonsingular cubic curve in this form is called an elliptic curve; for some authors, elliptic curves must have rational number coefficients. These include the Mordell curves ${\displaystyle y^{2}=x^{3}+n}$ and the Frey curves ${\displaystyle y^{2}=x(x-\alpha )(x+\beta )}$. Elliptic curves are commonly studied in number theory, in terms of the points on them that have rational coordinates, or more generally replacing the rationals by another number field.^[3] Elliptic curves over finite fields are commonly used in public key cryptography.

Through given points

Two cubic curves through the nine points of a 3 × 3 grid

A cubic curve F in the projective plane can be expressed as a non-zero linear combination of the third-degree monomials

⁠${\displaystyle x^{3},y^{3},z^{3},x^{2}y,x^{2}z,y^{2}x,y^{2}z,z^{2}x,z^{2}y,xyz}$⁠

These are ten in number; therefore the cubic curves themselves form a projective space of dimension 9, over any given field K. Each point P imposes a single linear condition on F, if we ask that C pass through P. Therefore, we can find some cubic curve through any nine given points. This cubic may be degenerate, and may not be unique, but will be unique and non-degenerate if the points are in general position; compare to two points determining a line and how five points determine a conic.

If two cubics pass through a given set of nine points, then in fact a pencil of infinitely many cubics does, and the points satisfy additional properties. Here, a pencil means a line in the nine-dimensional projective space of cubic curves. An example is given by the nine points of a ${\displaystyle 3\times 3}$ grid, which are passed through by two reducible cubics (three lines parallel to one of the grid axes) and therefore by an infinite family of cubics. According to the Cayley–Bacharach theorem, when this pencil of cubics exists, any cubic that passes through eight of the nine points must belong to the pencil and pass through all nine of them. This phenomenon forms the basis of Cramer's paradox: although nine points in general position are enough to determine a unique cubic curve through them, there exist sets of nine points (the nine points from a pencil of cubics) that do not determine a unique cubic.

Real points and inflections

The real points of cubic curves were studied by Isaac Newton. The real points of a non-singular projective cubic fall into one or two 'ovals'. One of these ovals crosses every real projective line, and thus is never bounded when the cubic is drawn in the Euclidean plane; it appears as one or three infinite branches, containing the three real inflection points. The other oval, if it exists, does not contain any real inflection point and appears either as an oval or as two infinite branches. Like for conic sections, a line cuts this oval at, at most, two points. For an elliptic curve ${\displaystyle y^{2}=x^{3}+ax^{2}+bx+c}$, there are one or two ovals accordingly as the cubic polynomial in ${\displaystyle x}$ has one or three real roots; the roots of the polynomial mark the crossing points of the ovals by the ${\displaystyle x}$-axis.

The Hesse configuration of nine complex inflection points of a non-singular cubic curve and the 12 lines through triples of these points, drawn as an abstract incidence structure. These points and lines cannot be arranged in the Euclidean plane.

A non-singular cubic curve is known to have nine points of inflection in the projective plane over an algebraically closed field such as the complex numbers. This can be shown by taking the homogeneous version of the Hessian matrix, which defines again a cubic, and intersecting it with C; the intersections are then counted by Bézout's theorem. However, only three of these points may be real, so that the others cannot be seen in the real projective plane by drawing the curve. In the case of the witch of Agnesi, one of the three real inflection points is infinite, so there are only two finite real inflection points. The nine inflection points of a non-singular cubic have the property that every line passing through two of them contains exactly three inflection points. Any such set of nine inflection points, and the 12 lines through triples of them, forms a copy of the Hesse configuration. The cubic curves having these nine points as their inflection points form a pencil. An example of a pencil of this type, having inflection points as its nine shared points, is the Hesse pencil of curves of the form ${\displaystyle \lambda (x^{3}+y^{3}+z^{3})=\mu xyz}$.^[4]

Projective classification

As discussed above, the cubic curves in a projective plane over any field form a nine-dimensional projective space. However, the space of projective transformations of the plane containing these curves has only eight degrees of freedom (the nine coefficients of a linear transformation on homogenous coordinates, minus one for scalar equivalences), so there is a one-dimensional family of cubic curves that are inequivalent under projective transformations.

Any non-singular cubic curve (over a field of characteristic ${\displaystyle \notin \{2,3\}}$) can be transformed by projective transformation into either of two canonical forms, the Hesse normal form ${\displaystyle x^{3}+y^{3}+z^{3}=3kxyz}$ (for a single coefficient ${\displaystyle k}$, a curve in the Hesse pencil), or the standard normal form or Weierstrass normal form ${\displaystyle y^{2}=x^{3}+ax+b}$ (for two coefficients ${\displaystyle a}$ and ${\displaystyle b}$). Every non-singular cubic curve can be placed into Hesse form,^[5] and every irreducible cubic curve with an inflection point can be placed into standard normal form, with the inflection point at infinity.^[6]

In the real case, the non-singular cubics are completely classified by the real coefficient ${\displaystyle k}$ of the Hesse normal form. Curves in this form have an isolated point when ${\displaystyle k=1}$, and are nonsingular when ${\displaystyle k\neq 1}$; in the limit as ${\displaystyle k\to \infty }$ they degenerate to a reducible cubic with three lines.^[7] Real curves with ${\displaystyle k>1}$ have two projective components and with ${\displaystyle k<1}$ they have one component. Two non-singular cubics are projectively equivalent if and only if they have the same Hesse normal form.^[8] The same curves are almost completely classified by the j-invariant of the standard normal form,^[9] $${\displaystyle J={\frac {4a^{3}}{4a^{3}+27b^{2}}}=\left({\frac {k^{4}+8k}{4k^{3}-4}}\right)^{3},}$$ a number that remains unchanged between projectively equivalent curves in different standard normal forms: each real number is the j-invariant of two different non-singular real cubic curves. These curves differ from each other in the sign of ${\displaystyle b}$ or, if ${\displaystyle b=0}$, in the sign of ${\displaystyle a}$. They are equivalent under complex projective transformations, but not under real projective transformations.^[10]

In the complex case, the non-singular cubics are completely classified by the j-invariant: every complex number is the j-invariant of a cubic curve, and two non-singular cubic curves are projectively equivalent if and only if they have the same j-invariant.^[11] The case for the coefficient ${\displaystyle k}$ of the Hesse normal form is more complicated. The curve is singular when ${\displaystyle k^{3}=1}$, or for the reducible cubic ${\displaystyle xyz=0}$ which can be interpreted as the Hesse normal form with ${\displaystyle k=\infty }$.^[12] There is a twelve-element finite group of Möbius transformations such that two curves in Hesse normal form, with coefficients ${\displaystyle k}$ and ${\displaystyle k'}$, are projectively equivalent if and only if some element of this group maps ${\displaystyle k}$ to ${\displaystyle k'}$. This same group can be used to provide a product formula mapping any coefficient ${\displaystyle k}$ to the corresponding j-invariant.^[11]

It follows from the symmetries of the Hesse normal form that every non-singular complex projective curve has a group of at least 18 projective automorphisms (projective transformations that leave the curve unchanged),^[13] and that every non-singular real projective curve has a group of at least 6 projective automorphisms.^[14]

Group of the points

See also: Elliptic curve § Group law

A remarkable property of the non singular cubic plane curve is that, if one of the inflection points is selected, the points of the curve form an abelian group. For an elliptic curve in the form ⁠${\displaystyle y^{2}=ax^{3}+bx^{2}+cx+d}$⁠, the chosen inflection point is conventionally the point at infinity in the direction of the ⁠${\displaystyle y}$⁠-axis (point of projective coordinates ⁠${\displaystyle x=0,y=1,t=0}$⁠).

The group law is defined as follow: the identity element ⁠${\displaystyle O}$⁠ is the chosen inflection point. For every point ⁠${\displaystyle P}$⁠, the additive inverse ⁠${\displaystyle -P}$⁠ is the third intersection point of the curve and the line passing through ⁠${\displaystyle O}$⁠ and ⁠${\displaystyle P}$⁠. Given two points ⁠${\displaystyle P}$⁠ and ⁠${\displaystyle Q}$⁠, their sum ⁠${\displaystyle P+Q}$⁠ is the additive inverse of the third intersection point of the curve and the line passing through ⁠${\displaystyle P}$⁠ and ⁠${\displaystyle Q}$⁠. In what preceeds, the line passing through two equal points of the curve is the tangent to the curve at the point, and if a line is tangent to the curve, two of the intersection points are equal (three, if the point is an inflection point).

The fact that the group law is defined by collinearity has important consequences. In particular, if the cubic curve is defined over a field ⁠${\displaystyle k}$⁠ (the coefficients of the curve equation belong to ⁠${\displaystyle k}$⁠), the ⁠${\displaystyle k}$⁠-rational points (points whose all coordinates belong to ⁠${\displaystyle k}$⁠) form a group for the above definition. So, there is a group of the ⁠${\displaystyle k}$⁠-rational points for every field containing the coefficients.

Associated with triangles

Main article: Catalogue of Triangle Cubics

The Neuberg cubic passing through 21 special points of a triangle

Relative to a given triangle, many named cubics pass through the vertices of the triangle and its triangle centers. These include the curves listed below using barycentric coordinates. In this coordinate system, each of the three coordinates ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ gives the signed distance from the line through one side of the triangle, normalized so that the vertices of the triangle have coordinates (0,0,1), (0,1,0), and (1,0,0). The examples below simplify the equations for each cubic using the cyclic sum notation $${\displaystyle {\begin{aligned}\sum _{\text{cyclic}}&f(x,y,z,a,b,c)=\\&f(a,b,c,x,y,z)+f(b,c,a,y,z,x)+f(c,a,b,z,x,y).\\\end{aligned}}}$$

Notable triangle cubics include the following.

• The Neuberg cubic has the equation^[15] $${\displaystyle \sum _{\text{cyclic}}[a^{2}(b^{2}+c^{2})-(b^{2}-c^{2})^{2}-2a^{4}]x(c^{2}y^{2}-b^{2}z^{2})=0.}$$
• The Thomson cubic has the equation^[16] $${\displaystyle \sum _{\text{cyclic}}x(c^{2}y^{2}-b^{2}z^{2})=0.}$$
• The McCay cubic has the equation^[17] $${\displaystyle \sum _{\text{cyclic}}a^{2}(b^{2}+c^{2}-a^{2})x(c^{2}y^{2}-b^{2}z^{2})=0.}$$

See also

• Twisted cubic, a cubic space curve