Cubic plane curve

"Cubic curve" redirects here. For the graph of a polynomial function of degree 3, see Cubic function.

Singular cubic y² = x² (x + 1).

A selection of cubic curves

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 ${\displaystyle F(X,Y,Y)}$ or by the corresponding polynomial in two variables ${\displaystyle f(x,y)=F(X,Y,1).}$ Starting from ⁠${\displaystyle f}$⁠, one can recover ⁠${\displaystyle F}$⁠ as ⁠${\displaystyle F(X,Y,Z)=Z^{3}f(X/Z,(Y/Z)}$⁠.

Typically, the coefficients of the polynomial belong to ${\displaystyle \mathbb {R} ,}$ but they may belong to any field ⁠${\displaystyle k}$⁠, in which case, one talks of a cubic defined over ⁠${\displaystyle k}$⁠. The points of the cubic are the points of the projective space of dimension three over the field of the complex numbers (or over an algebraic closure of ⁠${\displaystyle k}$⁠), whose projective coordinates satisfy the equation of the cubic $${\displaystyle F(X,Y,Z)=0.}$$ A point at infinity of the cubic is a point such that ⁠${\displaystyle Z=0}$⁠. A real point of the cubic is a point with real coordinates. A point defined over ⁠${\displaystyle k}$⁠ is a point with coordinates in ⁠${\displaystyle k}$⁠.

Generally, the defining polynomial is implicitly assumed to be irreducible, since, otherwise, the equation defines either three lines (not necessarily distinct), 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 (or degenerated 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: $${\displaystyle f(x,y)=a_{0}+a_{1}x+a_{2}y+a_{3}x^{2}+a_{4}xy+a_{5}y^{2}+a_{6}x^{3}+a_{7}x^{2}y+a_{8}xy^{2}+a_{9}y^{3}}$$

Typically, the coefficients ⁠${\displaystyle a_{i}}$⁠are real numbers, and the points of the cubic are real zeros of ⁠${\displaystyle f}$⁠. The nonreal complex zeros of ⁠${\displaystyle f}$⁠ 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 ⁠${\displaystyle f}$⁠ to the projective plane, by considering as points of the cubic the points of the projective plane whose projective coordinates satisfy ⁠${\displaystyle F(Z,Y,Z)=0}$⁠, where $${\displaystyle F(X,Y,Z)=a_{0}Z^{3}+a_{1}XZ^{2}+a_{2}YZ^{2}+a_{3}X^{2}Z+a_{4}XYZ+a_{5}Y^{2}Z+a_{6}X^{3}+a_{7}X^{2}Y+a_{8}XY^{2}+a_{9}Y^{3}.}$$ The points of the Euclidean plane are identified with the points of the projective plane with ⁠${\displaystyle Z\neq 0}$⁠ by the relation ⁠${\displaystyle x=X/Z,y=Y/Z}$⁠. The points of the cubic such that ⁠${\displaystyle Z=0}$⁠ are called the points at infinity of the cubic.

Everything that precedes applies by replacing the field ⁠${\displaystyle \mathbb {R} }$⁠ of the real numbers with any field ⁠${\displaystyle k}$⁠, the Euclidean plane with an affine plane over ⁠${\displaystyle k}$⁠,, the complex numbers with an algebraically closed field ⁠${\displaystyle K}$⁠ containing ⁠${\displaystyle k}$⁠, "real point" with "point defined over ⁠${\displaystyle k}$⁠" or "⁠${\displaystyle k}$⁠-point", etc.

A cubic is degenerated or decomposed if the polynomial ⁠${\displaystyle f}$⁠ (equivalently ⁠${\displaystyle F}$⁠) 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 ⁠${\displaystyle K}$⁠, every line intersects the conic in three points, not necessarily distinct (an exception occurs if the line is a component of a decomposed cubic).

Tangents and singular points

The equation of the tangent at a point of projective coordinates ⁠${\displaystyle X_{0},Y_{0},Z_{0}}$⁠ on the cubic is $${\displaystyle XF_{X}'(X_{0},Y_{0},Z_{0})+YF'_{Y}(X_{0},Y_{0},Z_{0})+ZF'_{Z}(X_{0},Y_{0},Z_{0})=0.}$$ If all three partial derivatives at ⁠${\displaystyle (X_{0},Y_{0},Z_{0})}$⁠ 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 (counting multiplicities, which are at least 2 for singular 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 ⁠${\displaystyle (X_{0}:Y_{0}:Z_{0})}$⁠,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.

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 ⁠${\displaystyle y=ax^{3}+bx^{2}+cx+d}$⁠ is regular in the Euclidean plane but has a singular point at infinity in the direction of the y-axis (the point of projective coordinates ⁠${\displaystyle (0:1:0)}$⁠). 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 ⁠${\displaystyle yp(x)=q(x)}$⁠, where ⁠${\displaystyle p}$⁠ and ⁠${\displaystyle q}$⁠ are polynomials in ⁠${\displaystyle x}$⁠ such that ⁠${\displaystyle \max(\deg(q),1+\deg(p))=3}$⁠.

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 thecissoid of Diocles. The curve ${\displaystyle y^{2}=x^{3}-x^{2}}$ is an example having an isolated point, at the origin.^[1]

Singular cubics are also called unicursal cubics, because a moving point travelling the cubic can cover the whole cubic in a single course (except, the isolated double point,if there is one). 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 their slope in the Euclidean plane. The three intersection points of 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 ${\displaystyle c(x,y)-q(x,y)=0,}$ where ⁠${\displaystyle c}$⁠ and ⁠${\displaystyle q}$⁠ are homogeneous polynomials of respective degrees 3 and 2. Setting ⁠${\displaystyle y=tx}$⁠, one gets $${\displaystyle c(x,tx)-q(x,y)=x^{2}(xc(1,t)-q(1,t)),}$$ giving the parametric equation $${\displaystyle x={\frac {q(1,t)}{c(1,t)}},\qquad y={\frac {tq(1,t)}{c(1,t)}}.}$$ If desired, one can make back the change of coordinates for having the parametrization in terms of the original coordinates.

Conversely, if ⁠${\displaystyle q_{0}(t)}$⁠, ⁠${\displaystyle q_{1}(t)}$⁠, and ⁠${\displaystyle q_{1}(t)}$⁠ are three polynomials without a common factor, that have 3 as their maximal degree, then the parametic equation $${\displaystyle x={\frac {q_{1}(t)}{q_{0}(t)}},\qquad y={\frac {q_{2}(t)}{q_{0}(t))}}}$$ defines a singular cubic whose implicit equation can be obtained as the resultant $${\displaystyle Res_{t}(xq_{0}(t)-q_{1}(t),yq_{0}(t)-q_{2}(t)).}$$

Weierstrass normal form

Over a field of characteristic different from 3, every irreducible cubic can be transformed into the Weierstrass normal form $${\displaystyle y^{2}=x^{3}+ax^{2}+bx+c}$$ by by a projective transformation, or equivalently by a change of projective coordinates.^[2] The parameters ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ may belong to the field of definition of the cubic even if the projective transdormation 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 ⁠${\displaystyle y}$⁠-axis (that is the point ⁠${\displaystyle (0:1:0)}$⁠), 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 $${\displaystyle ax^{3}+bx^{2}+cx+d=ey^{2}+fxy+gy.}$$ One has ⁠${\displaystyle a\neq 0}$⁠, since, otherwise, the line at infinity would be a component of the curve. One has also ⁠${\displaystyle e\neq 0}$⁠, since otherwise, the point ⁠${\displaystyle (0:1:0)}$⁠ would be a singular point and thus not an inflexion point.

The transformation ⁠${\displaystyle (x\mapsto aex,\quad y\mapsto a^{2}ex)}$⁠ and the division of the whole equation by ⁠${\displaystyle a^{4}e^{2}}$⁠ allows supposing ⁠${\displaystyle a=e=1}$⁠. The transformation ⁠${\displaystyle y\mapsto y-(fx+g)/2}$⁠ gives ⁠${\displaystyle f=g=0}$⁠ (variant of completing the square). Finally, the transformation ⁠${\displaystyle x\mapsto x-b/3}$⁠ (depressing the cubic) gives the Weierstrass normal form.

The Weierstrass normal form is not unique since the transformation ⁠${\displaystyle (x\mapsto x/\lambda ^{2},\quad y\mapsto y/\lambda ^{3})}$⁠ and the multiplication of the whole equation by ⁠${\displaystyle \lambda ^{6}}$⁠ amounts to nultiply the coefficient of ⁠${\displaystyle x}$⁠ and the constant coefficient by ⁠${\displaystyle \lambda ^{4}}$⁠ and ⁠${\displaystyle \lambda ^{6}}$⁠ respectively.

The invariant theory (see below) 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 ⁠${\displaystyle \lambda }$⁠ for getting a Weierstrass normal form with coefficients in the field of definition of the cubic.

Invariant theory

Invariant theory is mainly concerned with the study of invariants of homogeneous polynomials, called forms in this context, under the action of the projective special linear group (PSL) on the variables. An invariant for the forms of degree ⁠${\displaystyle d}$⁠ in ⁠${\displaystyle n}$⁠ variables is a polynomial with integer coefficients whose indeterminates are the coefficients of a generic form, that is left invariant under the action of ⁠${\displaystyle \operatorname {PSL} (n)}$⁠ on the variables of the form. For, example, if ⁠${\displaystyle d=n=2}$⁠ (binary quadratic forms), the generic form is $${\displaystyle ax^{2}+bxy+cy^{2}}$$ and the discriminant ⁠${\displaystyle b^{2}-4ac}$⁠ is an invariant that is essentially unique, since all invariants are polynomials in the discriminant.

Here we are concerned with ternary cubic forms, that is, homogeneous polynomials of degree 3 in 3 variables. Invariants are thus polynomials in 10 variables. The invariants form a ring ⁠${\displaystyle \mathbb {Z} [S,T]}$⁠ where ⁠${\displaystyle S}$⁠ and ⁠${\displaystyle T}$⁠ are homogeneous polynomials in 10 variables of respective degrees 4 and 6. The invariant ⁠${\displaystyle 4S^{3}+27Q^{2}}$⁠, of degree 12 is called the discriminant of the cubic.

Given the Weierstrass normal form of an irreducible cubic, one can choose the above coefficient ⁠${\displaystyle \lambda }$⁠ for having the Weierstrass normal form $${\displaystyle y^{2}=x_{3}+g_{2}x+g-3,}$$ where ⁠${\displaystyle g_{2}}$⁠ and ⁠${\displaystyle g_{3}}$⁠ are the value of the invariants ⁠${\displaystyle S}$⁠ and ⁠${\displaystyle T}$⁠ at the coefficients of the original cubic.

This shows that the Weierstrass normal form does not depends of the choice of an inflexion point and its coefficients may always be chosen in the field ⁠${\displaystyle k}$⁠ generated by the coefficients of the cubic. However if there is no inflexion point defined on ⁠${\displaystyle k}$⁠, the ⁠${\displaystyle k}$⁠-points of the Weierstrass normal form may be not the same as those of the original cubic.

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 ${\displaystyle \lambda (x^{3}+y^{3}+z^{3})=\mu xyz}$.^[3]

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,^[4] and every irreducible cubic curve with an inflection point can be placed into standard normal form, with the inflection point at infinity.^[5]

Curves in the Hesse pencil of cubic curves ${\displaystyle x^{3}+y^{3}+z^{3}=3kxyz}$, as seen in a symmetric view of the projective plane in which the lines ${\displaystyle x,y,z=0}$ form an equilateral triangle with the point 1:1:1 at its center, and the line ${\displaystyle x+y+z=0}$ forms the line at infinity.

Different colors represent different curves in the pencil, corresponding to different choices of the parameter ${\displaystyle k}$:

• The three black lines forming an equilateral triangle represent the degenerate case ${\displaystyle xyz=0}$ corresponding to ${\displaystyle k=\infty }$.
• The black point in the center is an isolated point from the degenerate case ${\displaystyle k=1}$. It has as its roots the point 1:1:1 (shown, from a quadratic factor) and the line at infinity (not shown, from a linear factor).
• The blue and green curves represent choices of ${\displaystyle k}$ with ${\displaystyle k>1}$, forming two ovals in the real projective plane: a central oval inside the equilateral triangle, and an external oval with three inflection points at infinity, separating it into three real branches. The choices of ${\displaystyle k}$ for the blue curves are (from lighter to darker) 36, 12, 6, 4, and 3. For the green curves, only the central oval is visible; their parameters are (from lighter to darker) 2, 1.5, 1.25, and 1.1
• The red curves represent choices of ${\displaystyle k}$ with ${\displaystyle k<1}$, having only a single oval with the same three inflection points and three real branches. Their choices of ${\displaystyle k}$ are (from lighter to darker) –8, –2, –1, –0.5, 0.

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 reduce to an isolated point and a line 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.^[6] 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.^[7] The same curves are almost completely classified by the j-invariant of the standard normal form,^[8] $${\displaystyle J={\frac {4a^{3}}{4a^{3}+27b^{2}}}=\left({\frac {k^{4}+8k}{4k^{3}-4}}\right)^{3},}$$ 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.^[9]

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.^[10] 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 ${\displaystyle k=\infty }$.^[11] 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.^[10]

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),^[12] and that every non-singular real projective curve has a group of at least 6 projective automorphisms.^[13]

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^[14] $${\displaystyle \sum _{\text{cyclic}}[a^{2}(b^{2}+c^{2})-(b^{2}-c^{2})^{2}-2a^{4}]x(c^{2}y^{2}-b^{2}z^{2})=0.}$$
• The Thomson cubic has the equation^[15] $${\displaystyle \sum _{\text{cyclic}}x(c^{2}y^{2}-b^{2}z^{2})=0.}$$
• The McCay cubic has the equation^[16] $${\displaystyle \sum _{\text{cyclic}}a^{2}(b^{2}+c^{2}-a^{2})x(c^{2}y^{2}-b^{2}z^{2})=0.}$$

See also

• Twisted cubic, a cubic space curve

Notes

1. Bix 1998, pp. 141–142.
2. Bix 1998, p. 127.
3. Artebani & Dolgachev 2009.
4. Bonifant & Milnor 2017, Theorem 2.12.
5. Bonifant & Milnor 2017, Theorem 3.1.
6. Bonifant & Milnor 2017, Fig. 10.
7. Bonifant & Milnor 2017, Theorem 6.3.
8. Bonifant & Milnor 2017, Equations (13) and (14).
9. Bonifant & Milnor 2017, Lemma 6.2.
10. Bonifant & Milnor 2017, Theorem 3.12.
11. Bonifant & Milnor 2017, Lemma 2.2.
12. Bonifant & Milnor 2017, Lemma 2.4.
13. Bonifant & Milnor 2017, Corollary 6.7.
14. Gibert, K001 Neuberg cubic
15. Gibert, K002 Thomson cubic
16. Gibert, K003 McCay cubic

References

• Artebani, Michela; Dolgachev, Igor (2009), "The Hesse pencil of plane cubic curves", L'Enseignement Mathématique, 55 (3–4): 235–273, arXiv:math/0611590, doi:10.4171/LEM/55-3-3, MR 2583779
• Bix, Robert (1998), Conics and Cubics: A Concrete Introduction to Algebraic Curves, New York: Springer, ISBN 0-387-98401-1
• Bonifant, Araceli; Milnor, John (2017), "On real and complex cubic curves", L'Enseignement Mathématique, 63 (1–2): 21–61, arXiv:1603.09018, doi:10.4171/LEM/63-1/2-2, MR 3777131
• Bernard, Gibert, "Catalogue of Triangle Cubics", Cubics in the Triangle Plane, Bernard Gibert, retrieved 2025-10-13

External links

• A Catalog of Cubic Plane Curves (archived version)
• Points on Cubics
• Cubics in the Triangle Plane
• Special Isocubics in the Triangle Plane (pdf), by Jean-Pierre Ehrmann and Bernard Gibert
• "Real and Complex Cubic Curves - John Milnor, Stony Brook University [2016]", YouTube, Graduate Mathematics, June 27, 2018 lecture in July 2016, ICMS, Edinburgh at conference in honour of Dusa McDuff's 70th birthday