CUSPIDAL CUBICS.[2]
Inverting the parabola from its vertex we obtain the Cissoid of Diocles. The focus of the parabola inverts into a point on the cuspidal tangent which I shall call the focus of the cissoid. The circle of curvature at the vertex of the parabola inverts into the asymptote of the cissoid. This asymptote is also plainly the inflectional tangent, and the point at infinity is the point of inflection. The directrix of the parabola inverts into a circle through the cusp of the cissoid having the cuspidal tangent for a diameter. Hall calls this the directrix circle. The double ordinate of the parabola which is tangent to the circle of curvature of the vertex inverts into the circle usually called the base circle of the cissoid.[3]
The cissoid may fairly be called the simplest form of the cuspidal cubic. Its projection and polar reciprocal are both cuspidal cubics. I shall now deduce from the parabola a few simple propositions for the cissoid, and then extend them to all cuspidal cubics.
(1) It is known that the locus of the intersection of tangents to the parabola which are at right angles to one another, is the directrix. Inverting:—the locus of the intersection of tangent circles to the cissoid through the cusp and at right angles to each other is the directrix circle.
(2) For the parabola, two right lines O P and O Q, are drawn through the vertex of the parabola at right angles to one another, meeting the curve in P and Q; the line P Q cuts the axis at a fixed point, whose abscissa is equal to its ordinate. Inverting:—two right lines, O P and O Q, are drawn at right angles to one another through the cusp of the cissoid, meeting the curve in P and Q; the circle O P Q passes through the intersection of the axis and asymptote.
(3) If the normals at the points P, O, R, of a parabola meet at a point, the circle through P O R will pass through the vertex. Inverting:—through a fixed point and the cusp of a cissoid, three and only three circles can be passed, cutting the cissoid at right angles; these three points of intersection are collinear.
From the geometry of the cissoid we see that if any line be drawn parallel to the asymptote, cutting the curve in two points, B and C, the segment B C is bisected by the axis. Hence, projecting the curve we have the following theorem:—any line drawn through the point of inflection is cut harmonically by the point of inflection, the curve, and the cuspidal tangent. Thus the cuspidal tangent is the harmonic polar of the point of inflection. The polar reciprocal of this last theorem reads as follows:—if from any point on the cuspidal tangent the two other tangent lines be drawn to the curve, and a line to the point of inflection, these four lines form a harmonic pencil. These are fundamental propositions in the theory of cuspidal cubics.
(4) Projecting proposition (1) above, we have the generalized theorem:—through the point of inflection draw any line cutting the cubic in B and C; through B, C, and the cusp draw two conics tangent to the cubic, and intersecting in a fourth point such that the two tangents to the conics at their point of intersection, together with the two lines from it to B and C, form a harmonic pencil; the locus of all such intersections is a conic through B, C, and the cusp having the point of inflection and the cuspidal tangent for pole and polar.
(5) Reciprocating (4) we have:—through any point on the cuspidal tangent draw the two other tangents, B and C, to the cubic. Touching B, C, and the inflectional tangent draw two conics, such that the points of contact of their common tangent, together with the points where their common tangent cuts the tangents B and C, form a harmonic range; the envelope of such common tangents is a conic having the cuspidal tangent and the point of inflection for polar and pole.
(6) Projecting (2) we obtain the following:—through the point of inflection draw any line cutting the curve in B and C; take any other two points on the cubic such that the pencil from the cusp, O, O (B P C Q) is harmonic; the conic passing through O B P C Q will pass through the intersection of the cuspidal and inflectional tangents.