STROPHOIDS, A FAMILY OF CUBIC CURVES WITH REMARKABLE PROPERTIES

Abstract

Strophoids are circular cubic curves which have a node with orthogonal tangents. These rational curves are characterized by a series or properties, and they show up as locus of points at various geometric problems in the Euclidean plane: Strophoids are pedal curves of parabolas if the corresponding pole lies on the parabola’s directrix, and they are inverse to equilateral hyperbolas. Strophoids are focal curves of particular pencils of conics. Moreover, the locus of points where tangents through a given point contact the conics of a confocal family is a strophoid. In descriptive geometry, strophoids appear as perspective views of particular curves of intersection, e.g., of Viviani’s curve. Bricard’s flexible octahedra of type 3 admit two flat poses; and here, after fixing two opposite vertices, strophoids are the locus for the four remaining vertices. In plane kinematics they are the circle-point curves, i.e., the locus of points whose trajectories have instantaneously a stationary curvature. Moreover, they are projections of the spherical and hyperbolic analogues. For any given triangle ABC, the equicevian cubics are strophoids, i.e., the locus of points for which two of the three cevians have the same lengths. On each strophoid there is a symmetric relation of points, so-called ‘associated’ points, with a series of properties: The lines connecting associated points P and P’ are tangent of the negative pedal curve. Tangents at associated points intersect at a point which again lies on the cubic. For all pairs (P, P’) of associated points, the midpoints lie on a line through the node N. For any two pairs (P, P’) and (Q, Q’) of associated points, the points of intersection between the lines PQ and P’Q’ as well as between PQ’ and P’Q are again placed on the strophoid and mutually associated. The lines PQ and PQ’ are symmetric with respect to the line connecting P with the node. Thus, strophoids generalize Apollonian circles: For given non-collinear points A, A’ and N the locus of points X such that one angle bisector of the lines XA and XA’ passes through N is a strophoid.