Cubic plane curve

File:CubicCurve.svg

A selection of cubic curves.

In mathematics, a cubic plane curve is a plane algebraic curve Template:Mvar defined by a cubic equation

Template:Tmath

applied to homogeneous coordinates Template:Tmath for the projective plane; or the inhomogeneous version for the affine space determined by setting Template:Math in such an equation. Here Template:Mvar is a non-zero linear combination of the third-degree monomials

Template:Tmath

These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given field Template:Mvar. Each point Template:Mvar imposes a single linear condition on Template:Mvar, if we ask that Template:Mvar pass through Template:Mvar. Therefore, we can find some cubic curve through any nine given points, which 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 cubics does, and the points satisfy additional properties; see Cayley–Bacharach theorem.

File:Cubic with double point.svg

Singular cubic Template:Math. A parametrization is given by Template:Math.

A cubic curve may have a singular point, in which case it has a parametrization in terms of a projective line. Otherwise a non-singular cubic curve is known to have nine points of inflection, 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 Template:Mvar; 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. 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.

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.

A non-singular plane cubic defines an elliptic curve, over any field Template:Mvar for which it has a point defined. Elliptic curves are now normally studied in some variant of Weierstrass's elliptic functions, defining a quadratic extension of the field of rational functions made by extracting the square root of a cubic. This does depend on having a Template:Mvar-rational point, which serves as the point at infinity in Weierstrass form. There are many cubic curves that have no such point, for example when Template:Mvar is the rational number field.

The singular points of an irreducible plane cubic curve are quite limited: one double point, or one cusp. A reducible plane cubic curve is either a conic and a line or three lines, and accordingly have two double points or a tacnode (if a conic and a line), or up to three double points or a single triple point (concurrent lines) if three lines.

Cubic curves in the plane of a triangle

Suppose that Template:Math is a triangle with sidelengths <math>a = |BC|,</math> <math>b = |CA|,</math> <math>c = |AB|.</math> Relative to Template:Math, many named cubics pass through well-known points. Examples shown below use two kinds of homogeneous coordinates: trilinear and barycentric.

To convert from trilinear to barycentric in a cubic equation, substitute as follows:

<math>x \to bcx, \quad y \to cay, \quad z \to abz;</math>

to convert from barycentric to trilinear, use

<math>x \to ax, \quad y \to by, \quad z \to cz.</math>

Many equations for cubics have the form

<math>f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,z,x,y) = 0.</math>

In the examples below, such equations are written more succinctly in "cyclic sum notation", like this:

<math>\sum_{\text{cyclic}} f(x,y,z,a,b,c) = 0 </math>.

The cubics listed below can be defined in terms of the isogonal conjugate, denoted by Template:Mvar, of a point Template:Mvar not on a sideline of Template:Math. A construction of Template:Mvar follows. Let Template:Mvar be the reflection of line Template:Mvar about the internal angle bisector of angle Template:Mvar, and define Template:Mvar and Template:Mvar analogously. Then the three reflected lines concur in Template:Mvar. In trilinear coordinates, if <math>X = x:y:z,</math> then <math>X^* = \tfrac{1}{x}:\tfrac{1}{y}:\tfrac{1}{z}.</math>

Neuberg cubic

File:NeubergCurve.png

Neuberg cubic of triangle Template:Math: The locus of Template:Mvar such that, if Template:Mvar are the reflections of Template:Mvar in the sidelines Template:Mvar, then the lines Template:Mvar are concurrent.

Trilinear equation: <math>\sum_{\text{cyclic}} (\cos{A} - 2\cos{B}\cos{C})x(y^2-z^2)= 0 </math>

Barycentric equation: <math>\sum_{\text{cyclic}} (a^2(b^2 + c^2) + (b^2 - c^2)^2 - 2a^4)x(c^2y^2-b^2z^2) = 0 </math>

The Neuberg cubic (named after Joseph Jean Baptiste Neuberg) is the locus of a point Template:Mvar such that Template:Mvar is on the line Template:Mvar, where Template:Mvar is the Euler infinity point (Template:Math in the Encyclopedia of Triangle Centers). Also, this cubic is the locus of Template:Mvar such that the triangle Template:Math is perspective to Template:Math, where Template:Math is the reflection of Template:Mvar in the lines Template:Mvar respectively

The Neuberg cubic passes through the following points: incenter, circumcenter, orthocenter, both Fermat points, both isodynamic points, the Euler infinity point, other triangle centers, the excenters, the reflections of Template:Mvar in the sidelines of Template:Math, and the vertices of the six equilateral triangles erected on the sides of Template:Math.

For a graphical representation and extensive list of properties of the Neuberg cubic, see K001 at Berhard Gibert's Cubics in the Triangle Plane.

Thomson cubic

File:Thomson cubic.svg

Example of Thomson cubic (black curve). Template:Mvar is on the cubic, such that the isogonal conjugate of Template:Math is on the line Template:Math.

Trilinear equation: <math>\sum_{\text{cyclic}} bcx(y^2-z^2)= 0 </math>

Barycentric equation: <math>\sum_{\text{cyclic}} x(c^2y^2-b^2z^2)= 0 </math>

The Thomson cubic is the locus of a point Template:Mvar such that Template:Mvar is on the line Template:Mvar, where Template:Mvar is the centroid.

The Thomson cubic passes through the following points: incenter, centroid, circumcenter, orthocenter, symmedian point, other triangle centers, the vertices Template:Mvar the excenters, the midpoints of sides Template:Mvar and the midpoints of the altitudes of Template:Math. For each point Template:Mvar on the cubic but not on a sideline of the cubic, the isogonal conjugate of Template:Mvar is also on the cubic.

For graphs and properties, see K002 at Cubics in the Triangle Plane.

Darboux cubic

File:DarbouxCubic.png

Darboux cubic of triangle Template:Math: The locus of Template:Mvar such that if Template:Mvar are the feet of perpendiculars from Template:Mvar to the sidelines Template:Mvar then the lines Template:Mvar are concurrent.

Trilinear equation:<math>\sum_{\text{cyclic}} (\cos{A} - \cos{B}\cos{C})x(y^2-z^2)= 0 </math>

Barycentric equation: <math>\sum_{\text{cyclic}} (2a^2(b^2 + c^2) + (b^2 - c^2)^2 - 3a^4)x(c^2y^2-b^2z^2) = 0 </math>

The Darboux cubic is the locus of a point Template:Mvar such that Template:Mvar is on the line Template:Mvar, where Template:Mvar is the de Longchamps point. Also, this cubic is the locus of Template:Mvar such that the pedal triangle of Template:Mvar is the cevian triangle of some point (which lies on the Lucas cubic). Also, this cubic is the locus of a point Template:Mvar such that the pedal triangle of Template:Mvar and the anticevian triangle of Template:Mvar are perspective; the perspector lies on the Thomson cubic.

The Darboux cubic passes through the incenter, circumcenter, orthocenter, de Longchamps point, other triangle centers, the vertices Template:Mvar the excenters, and the antipodes of Template:Mvar on the circumcircle. For each point Template:Mvar on the cubic but not on a sideline of the cubic, the isogonal conjugate of Template:Mvar is also on the cubic.

For graphics and properties, see K004 at Cubics in the Triangle Plane.

Napoleon–Feuerbach cubic

Trilinear equation: <math>\sum_{\text{cyclic}} \cos(B-C)x(y^2-z^2)= 0 </math>

Barycentric equation: <math>\sum_{\text{cyclic}} (a^2(b^2 + c^2) + (b^2 - c^2)^2)x(c^2y^2-b^2z^2) = 0 </math>

The Napoleon–Feuerbach cubic is the locus of a point Template:Mvar is on the line Template:Mvar, where Template:Mvar is the nine-point center, (Template:Math in the Encyclopedia of Triangle Centers).

The Napoleon–Feuerbach cubic passes through the incenter, circumcenter, orthocenter, 1st and 2nd Napoleon points, other triangle centers, the vertices Template:Mvar the excenters, the projections of the centroid on the altitudes, and the centers of the 6 equilateral triangles erected on the sides of Template:Math.

For a graphics and properties, see K005 at Cubics in the Triangle Plane.

Lucas cubic

File:LucasCubic.png

Lucas Cubic of triangle Template:Math: The locus of a point Template:Mvar such that the cevian triangle of Template:Mvar is the pedal triangle of some point Template:Mvar; the point Template:Mvar lies on the Darboux cubic.

Trilinear equation: <math>\sum_{\text{cyclic}} \cos(A)x(b^2y^2- c^2z^2)= 0 </math>

Barycentric equation: <math>\sum_{\text{cyclic}} (b^2+c^2-a^2)x(y^2-z^2)= 0 </math>

The Lucas cubic is the locus of a point Template:Mvar such that the cevian triangle of Template:Mvar is the pedal triangle of some point; the point lies on the Darboux cubic.

The Lucas cubic passes through the centroid, orthocenter, Gergonne point, Nagel point, de Longchamps point, other triangle centers, the vertices of the anticomplementary triangle, and the foci of the Steiner circumellipse.

For graphics and properties, see K007 at Cubics in the Triangle Plane.

1st Brocard cubic

File:FirstBrocardCubic.png

First Brocard Cubic: It is the locus of Template:Mvar such the intersections of Template:Mvar with the sidelines Template:Mvar where Template:Math is the first Brocard triangle of triangle Template:Math, are collinear. In the figure Template:Mvar and Template:Mvar are the first and second Brocard points.

Trilinear equation:<math>\sum_{\text{cyclic}} bc(a^4-b^2c^2)x(y^2+z^2)= 0 </math>

Barycentric equation: <math>\sum_{\text{cyclic}} (a^4-b^2c^2)x(c^2y^2+b^2z^2)= 0 </math>

Let Template:Math be the 1st Brocard triangle. For arbitrary point Template:Mvar, let Template:Mvar be the intersections of the lines Template:Mvar with the sidelines Template:Mvar respectively. The 1st Brocard cubic is the locus of Template:Mvar for which the points Template:Mvar are collinear.

The 1st Brocard cubic passes through the centroid, symmedian point, Steiner point, other triangle centers, and the vertices of the 1st and 3rd Brocard triangles.

For graphics and properties, see K017 at Cubics in the Triangle Plane.

2nd Brocard cubic

Trilinear equation: <math>\sum_{\text{cyclic}} bc(b^2-c^2)x(y^2+z^2)= 0 </math>

Barycentric equation: <math>\sum_{\text{cyclic}} (b^2-c^2)x(c^2y^2+b^2z^2)= 0 </math>

The 2nd Brocard cubic is the locus of a point Template:Mvar for which the pole of the line Template:Mvar in the circumconic through Template:Mvar and Template:Mvar lies on the line of the circumcenter and the symmedian point (i.e., the Brocard axis). The cubic passes through the centroid, symmedian point, both Fermat points, both isodynamic points, the Parry point, other triangle centers, and the vertices of the 2nd and 4th Brocard triangles.

For a graphics and properties, see K018 at Cubics in the Triangle Plane.

1st equal areas cubic

File:FirstEqualAreasCubic.png

First equal area cubic of triangle Template:Math: The locus of a point Template:Mvar such that area of the cevian triangle of Template:Mvar equals the area of the cevian triangle of Template:Mvar.

Trilinear equation: <math>\sum_{\text{cyclic}} a(b^2-c^2)x(y^2-z^2)= 0 </math>

Barycentric equation: <math>\sum_{\text{cyclic}} a^2(b^2-c^2)x(c^2y^2-b^2z^2)= 0 </math>

The 1st equal areas cubic is the locus of a point Template:Mvar such that area of the cevian triangle of Template:Mvar equals the area of the cevian triangle of Template:Mvar. Also, this cubic is the locus of Template:Mvar for which Template:Mvar is on the line Template:Mvar, where Template:Mvar is the Steiner point. (Template:Math in the Encyclopedia of Triangle Centers).

The 1st equal areas cubic passes through the incenter, Steiner point, other triangle centers, the 1st and 2nd Brocard points, and the excenters.

For a graphics and properties, see K021 at Cubics in the Triangle Plane.

2nd equal areas cubic

Trilinear equation: <math>(bz+cx)(cx+ay)(ay+bz) = (bx+cy)(cy +az)(az+bx) </math>

Barycentric equation:<math>\sum_{\text{cyclic}} a(a^2-bc)x(c^3y^2 - b^3z^2) = 0 </math>

For any point <math>X = x:y:z</math> (trilinears), let <math>X_Y = y:z:x</math> and <math>X_Z = z:x:y.</math> The 2nd equal areas cubic is the locus of Template:Mvar such that the area of the cevian triangle of Template:Mvar equals the area of the cevian triangle of Template:Mvar.

The 2nd equal areas cubic passes through the incenter, centroid, symmedian point, and points in Encyclopedia of Triangle Centers indexed as X(31), X(105), X(238), X(292), X(365), X(672), X(1453), X(1931), X(2053), and others.

For a graphics and properties, see K155 at Cubics in the Triangle Plane.

See also

• Cayley–Bacharach theorem, on the intersection of two cubic plane curves
• Twisted cubic, a cubic space curve
• Elliptic curve
• Witch of Agnesi
• Catalogue of Triangle Cubics

References

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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
• {{#invoke:citation/CS1|citation

|CitationClass=web }} lecture in July 2016, ICMS, Edinburgh at conference in honour of Dusa McDuff's 70th birthday Template:Algebraic curves navbox