An elementary treatise on cubic and quartic curves, by A. B. Basset.

34

TANGENTIAL COORDINATES.

I. Let OY be the perpendicular on to the tangent at any point
P of a curve, and let OZ be the perpendicular from 0 on to the
tangent at Y to the locus of Y; then the angle OP Y = YZ; and
OP. OZ = O2.
II. Let OP be produced to Q so that OP. OQ = k2, where k is
a constant. Let the tangent at Q to the locus of Q meet the tangent
at P in T. Then the angle TPQ = TQP.
The locus of Y is the first positive pedal, and the locus of Q is
the inverse of the original curve. Also the reciprocal polar is the
inverse of the pedal. We can now prove that:59. A node corresponds to a double tangent on the reciprocal
polar, and vice versa.
Let NY, NY' be the tangents at a node N; from the origin
O draw OY, OY' perpendicular to NY, NY', and produce them
to Q, Q' so that
OY. OQ= OY'. OQ'= k............. (11).
Join QQ', YY' and draw TY such that the angle TYQ = TQY.
Q
N
From (11) it follows that a circle can be described through
QYY'Q'; also a circle can be described through OYNY'; whence
Q'QY= YY'O = ONY,
accordingly a circle can be described through NZYQ, and therefore
the angle NZQ is a right angle. Whence
OZ. ON= OY. OQ = k and TYQ = TQY= ONY,
and therefore TY is the tangent at Y to the pedal, and TQ is the
tangent at Q to the reciprocal polar.
Similarly TQ' is the tangent at Q', and therefore QQ' touches
the reciprocal polar at Q and Q'. Also since OZ. ON= c2, QQ' is
the polar of N.