Cubic plane curve

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

⁠ for the projective plane; or the inhomogeneous version for the affine space determined by setting z = 1 in such an equation.

Here F is a non-zero linear combination of the third-degree monomials These are ten in number; therefore the cubic curves 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, 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.

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 C; the intersections are then counted by Bézout's theorem.

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.

Like for conic sections, a line cuts this oval at, at most, two points.

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.

There are many cubic curves that have no such point, for example when K is the rational number field.

Relative to △ABC, 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: to convert from barycentric to trilinear, use Many equations for cubics have the form In the examples below, such equations are written more succinctly in "cyclic sum notation", like this: The cubics listed below can be defined in terms of the isogonal conjugate, denoted by X*, of a point X not on a sideline of △ABC.

The Neuberg cubic (named after Joseph Jean Baptiste Neuberg) is the locus of a point X such that X* is on the line EX, where E is the Euler infinity point (X(30) in the Encyclopedia of Triangle Centers).

Also, this cubic is the locus of X such that the triangle △XAXBXC is perspective to △ABC, where △XAXBXC is the reflection of X in the lines BC, CA, AB, 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 A, B, C in the sidelines of △ABC, and the vertices of the six equilateral triangles erected on the sides of △ABC.

The Thomson cubic is the locus of a point X such that X* is on the line GX, where G is the centroid.

The Thomson cubic passes through the following points: incenter, centroid, circumcenter, orthocenter, symmedian point, other triangle centers, the vertices A, B, C, the excenters, the midpoints of sides BC, CA, AB, and the midpoints of the altitudes of △ABC.

Also, this cubic is the locus of a point X such that the pedal triangle of X and the anticevian triangle of X 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 A, B, C, the excenters, and the antipodes of A, B, C on the circumcircle.

The Napoleon–Feuerbach cubic passes through the incenter, circumcenter, orthocenter, 1st and 2nd Napoleon points, other triangle centers, the vertices A, B, C, the excenters, the projections of the centroid on the altitudes, and the centers of the 6 equilateral triangles erected on the sides of △ABC.

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 arbitrary point X, let XA, XB, XC be the intersections of the lines XA′, XB′, XC′ with the sidelines BC, CA, AB, respectively.

The 1st Brocard cubic is the locus of X for which the points XA, XB, XC 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.

The 2nd Brocard cubic is the locus of a point X for which the pole of the line XX* in the circumconic through X and X* lies on the line of the circumcenter and the symmedian point (i.e., the Brocard axis).

Also, this cubic is the locus of X for which X* is on the line S*X, where S is the Steiner point.

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.

A selection of cubic curves. Click the image to see information page for details.
Singular cubic y 2 = x 2 ⋅ ( x + 1) . A parametrization is given by t ↦ ( t 2 – 1, t ⋅ ( t 2 – 1)) .
Neuberg cubic of triangle △ ABC : The locus of X such that, if X A , X B , X C are the reflections of X in the sidelines BC, CA, AB , then the lines AX A , BX B , CX C are concurrent.
Example of Thomson cubic (black curve). X is on the cubic, such that the isogonal conjugate of X ( X ′) is on the line X (2) – X .
Darboux cubic of triangle △ ABC : The locus of X such that if D, E, F are the feet of perpendiculars from X to the sidelines BC, CA, AB then the lines AD, BE, CF are concurrent.
Lucas Cubic of triangle △ ABC : The locus of a point X such that the cevian triangle of X is the pedal triangle of some point X' ; the point X' lies on the Darboux cubic.
First Brocard Cubic: It is the locus of X such the intersections of XA', XB', XC' with the sidelines BC, CA, CB, where △ A'B'C' is the first Brocard triangle of triangle △ ABC , are collinear. In the figure Ω and Ω′ are the first and second Brocard points.
First equal area cubic of triangle △ ABC : The locus of a point X such that area of the cevian triangle of X equals the area of the cevian triangle of X* .