Cubic

A plane curve of the third order, i.e. a set of points in the (projective, affine, Euclidean) plane the homogeneous coordinates $ x _ {0} , x _ {1} , x _ {2} $ of which (in a projective, affine or Cartesian coordinate system, respectively) satisfy a homogeneous equation of the third degree:

$$ F ( x) \equiv \ \sum _ {i, j, k = 0 } ^ { 2 } a _ {ijk} x _ {i} x _ {j} x _ {k} = 0,\ \ a _ {ijk} = a _ {jik} = a _ {ikj} . $$

The number of tangents that can be drawn to a cubic from a point outside it is known as the class of the cubic. The conic

$$ \sum _ {i = 0 } ^ { 2 } \frac{\partial F }{\partial x _ {i} } x _ {i} ^ \prime = 0 $$

is known as the conic (or first) polar of the point $ M ^ { \prime } ( x _ {0} ^ \prime , x _ {1} ^ \prime , x _ {2} ^ \prime ) $; the point itself is called a pole. The straight line

$$ \sum _ {i = 0 } ^ { 2 } \frac{\partial F }{\partial x _ {i} ^ \prime } x _ {i} = 0 $$

is known as the rectilinear (or second) polar of the point relative to the cubic. If the pole $ M ^ { \prime } $ is a point of the cubic, its rectilinear polar is tangent at $ M ^ { \prime } $ to the cubic itself and to the conic polar of $ M ^ { \prime } $. The Hessian of a cubic is the set of points whose conic polars consist of two straight lines; it is defined by the equation

$$ H _ {3} \equiv \ \mathop{\rm det} \left ( \frac{\partial ^ {2} F }{\partial x _ {i} \partial x _ {j} } \right ) = 0. $$

A cubic intersects its Hessian at nine common points of inflection. The straight lines into which the conic polars of the points of the Hessian split, and also the straight lines joining pairs of corresponding points of the Hessian, form the envelope of a curve of order six and of the third class — the Cayleyan of the cubic. The set of cubics on the plane that pass through the nine points of inflection of a given cubic forms a syzygetic pencil, which contains the Hessians of all curves in the pencil and four curves, each of which splits into three straight lines and forms a syzygetic triangle. The conic polar of a point of inflection $ M ^ { \prime } $ splits into two straight lines: The tangent to the cubic at $ M ^ { \prime } $ and the harmonic polar of $ M ^ { \prime } $— the set of points harmonically conjugate to $ M ^ { \prime } $ relative to the two points at which the cubic intersects the secant through $ M ^ { \prime } $. The harmonic polars of three collinear points of inflection intersect at a single point. There exist various projective, affine and metric classifications of cubics: according to the types of the canonical equations; according to the types of the singular points of the cubic; according to the nature of the asymptotes; etc.

The best-known cubics on the Euclidean plane are: the folium of Descartes ( $ x ^ {3} + y ^ {3} - 3axy = 0 $); the witch of Agnesi ( $ y ( a ^ {2} + x ^ {2} ) = a ^ {3} $); the cubic parabola ( $ y = ax ^ {3} $); the semi-cubic parabola ( $ y ^ {2} = ax ^ {3} $); the strophoid ( $ y ^ {2} ( a - x) = x ^ {2} ( a + x) $); the cissoid of Diocles ( $ y ^ {2} ( 2a - x) = x ^ {3} $); the trisectrix ( $ x ( x ^ {2} + y ^ {2} ) = a ( 3x ^ {2} - y ^ {2} ) $); and the conchoid of Sluze ( $ a ( x - a) ( x ^ {2} + y ^ {2} ) = k ^ {2} x ^ {2} $). In algebraic geometry, the term cubic is applied both to a cubic hypersurface and to a three-dimensional cubic curve.

References

[1]	A.S. Smogorzhevskii, E.S. Stolova, "Handbook of the theory of planar curves of the third order" , Moscow (1961) (In Russian)

References

[a1]	E. Brieskorn, H. Knörrer, "Plane algebraic curves" , Birkhäuser (1986) (Translated from German) MR0886476 Zbl 0588.14019