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.
(7) Reciprocating (6):—from any point on the cuspidal tangent draw two
other tangents, B and C, to the cubic; take any two other tangents, P
and Q, such that the range cut from the inflectional tangent by B, C,
P, Q, is harmonic; the conic touching B, C, P, Q, and the inflectional
tangent will also touch the line joining the point of inflection and
the cusp.
(8) Projecting (3):—through the point of inflection draw any line
cutting the cubic in B and C; through the cusp O and the points B and C
on the cubic and any other fixed point P, three, and only three, conics
can be passed, such that the tangent to the conic and cubic at their
remaining point of intersection, together with the lines from it to B
and C, form a harmonic pencil; these three points of intersection are
collinear.