In mathematics, a cubic plane curve is a plane
algebraic curve C defined by a cubic equation
F(x, y, z) = 0applied to homogeneous coordinates
x:y:z 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
x3, y3, z3, x2y, x2z, y2x, y2z, z2x, z2y,
xyz.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. 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 K 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 K-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 K 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 ABC is a triangle with sidelengths
a = |BC|, b = |CA|, c = |AB|. 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:
x ↦ bcx, y ↦ cay, z ↦ abz;to convert
from barycentric to trilinear, use
x ↦ ax, y ↦ by, z ↦ cz.Many equations
for cubics have the form
f(a, b, c, x, y, z) + f(b, c, a, y, z, x)
+ f(c, a, b, z, x, y) = 0.In the examples
below, such equations are written more succinctly
in "cyclic sum notation", like this:
[cyclic sum f(x, y, z, a, b, c)] = 0.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. A construction
of X* follows. Let LA be the reflection of
line XA about the internal angle bisector
of angle A, and define LB and LC analogously.
Then the three reflected lines concur in X*.
In trilinear coordinates, if X = x:y:z, then
X* = 1/x:1/y:1/z.
=== Neuberg cubic ===
Trilinear equation: [cyclic sum (cos A − 2
cos B cos C)x(y2 − z2)] = 0
Barycentric equation: [cyclic sum (a2(b2 +
c2) + (b2 − c2)2 − 2a4)x(c2y2 − b2z2)]
= 0
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.
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 ===
Trilinear equation: [cyclic sum bcx(y2 − z2)]
= 0
Barycentric equation: [cyclic sum x(c2y2 − b2z2)]
= 0
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. For each
point P on the cubic but not on a sideline
of the cubic, the isogonal conjugate of P
is also on the cubic.
For graphs and properties, see K002 at Cubics
in the Triangle Plane.
=== Darboux cubic ===
Trilinear equation: [cyclic sum (cos A − cos
B cos C)x(y2 − z2)] = 0
Barycentric equation: [cyclic sum (2a2(b2
+ c2) + (b2 − c2)2 − 3a4)x(c2y2 − b2z2)]
= 0
The Darboux cubic is the locus of a point
X such that X* is on the line LX, where L
is the de Longchamps point. Also, this cubic
is the locus of X such that the pedal triangle
of X is the cevian of some point (which lies
on the Lucas cubic). 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. For each point P
on the cubic but not on a sideline of the
cubic, the isogonal conjugate of P is also
on the cubic.
For graphics and properties, see K004 at Cubics
in the Triangle Plane.
=== Napoleon–Feuerbach cubic ===
Trilinear equation: [cyclic sum cos(B − C)x(y2
− z2)] = 0
Barycentric equation: [cyclic sum (a2(b2 +
c2) − (b2 − c2)2)x(c2y2 − b2z2)] = 0
The Napoleon–Feuerbach cubic is the locus
of a point X* is on the line NX, where N is
the nine-point center, (N = X(5) 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 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.
For a graphics and properties, see K005 at
Cubics in the Triangle Plane.
=== Lucas cubic ===
Trilinear equation: [cyclic sum (cos A)x(b2y2
− c2z2)] = 0
Barycentric equation: [cyclic sum (b2 + c2
− a2)x(y2 − z2)] = 0
The Lucas cubic is the locus of a point X
such that the cevian triangle of X 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 ===
Trilinear equation: [cyclic sum bc(a4 − b2c2)x(y2
+ z2] = 0
Barycentric equation: [cyclic sum (a4 − b2c2)x(c2y2
+ b2z2] = 0
Let A′B′C′ be the 1st Brocard triangle.
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.
For graphics and properties, see K017 at Cubics
in the Triangle Plane.
=== 2nd Brocard cubic ===
Trilinear equation: [cyclic sum bc(b2 − c2)x(y2
+ z2] = 0
Barycentric equation: [cyclic sum (b2 − c2)x(c2y2
+ b2z2] = 0
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).
The 2nd Brocard 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 ===
Trilinear equation: [cyclic sum a(b2 − c2)x(y2
− z2] = 0
Barycentric equation: [cyclic sum a2(b2 − c2)x(c2y2
− b2z2] = 0
The 1st equal areas cubic is the locus of
a point X such that area of the cevian triangle
of X equals the area of the cevian triangle
of X*. Also, this cubic is the locus of X
for which X* is on the line S*X, where S is
the Steiner point. (S = X(99) 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: (bz + cx)(cx + ay)(ay
+ bz) = (bx + cy)(cy + ax)(az + bx)
Barycentric equation: [cyclic sum a(a2 − bc)x(c3y2
− b3z2)] = 0
For any point X = x:y:z (trilinears), let
XY = y:z:x and XZ = z:x:y. The 2nd equal areas
cubic is the locus of X such that the area
of the cevian triangle of XY equals the area
of the cevian triangle of XZ.
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
