Geometry and Group Structures of Some Cubics

We review the group structure of a cubic in the projective complex plane and give group theoretic formulations of some geometric properties of a cubic. Then, we apply them to pivotal isocubics, in particular to the cubics of Thomson, Darboux and Lucas. We use the group structure to identify different transformations of cubics. We also characterize equivalence of cubics in terms of the Salmon cross ratio. 1. The group structure of a cubic Let Γ be a nonsingular cubic curve in the complex projective plane, i.e., Γ has no cusp and no node. It is well known that Γ has a group structure, which does not depend on the choice of a neutral element O on the cubic. In other words, the group structures on Γ for various choices of the neutral elements are isomorphic. If P and Q are points of a cubic Γ, we denote by P ·Q the third intersection of the line PQ with Γ. In particular, P · P := Pt is the tangential of P , the second intersection of Γ with the tangent at P . Proposition 1. The operation · is commutative but not associative. For P , Q, R on Γ, (1) (P ·Q) · P = Q, (2) P ·Q = R ·Q ⇐⇒ P = R, (3) P ·Q = R ⇐⇒ P = R ·Q. Convention: When we write P ·Q ·R, we mean (P ·Q) · R. We choose a point O on Γ as the neutral point, 1 and define a group structure + on Γ by P +Q = (P ·Q) ·O. We call the tangential of O, the point N = Ot = O ·O, the constant point of Γ. Note that −N = Nt, since N +Nt = N ·Nt ·O = N ·O = O. We begin with a fundamental result whose proof can be found in [4, p.392]. Theorem 2. 3k points Pi, 1 ≤ i ≤ 3k, of a cubic Γ are on a curve of order k if and only if ∑ Pi = kN . For k = 1, 2, 3, we have the following corollary. Corollary 3. Let P , Q, R, S, T , U , V , W , X be points of Γ. (1) P , Q, R are collinear if and only if P +Q+R = N . (2) P , Q, R, S, T , U are on a conic if and only if P+Q+R+S+T+U = 2N . Publication Date: November 8, 2002. Communicating Editor: Paul Yiu. 1O is not necessarily an inflexion point (a flex).