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

40 TANGENTIAL COORDINATES.
On the Curves r71 = an cos nO.
67. We have investigated several theorems concerning the
important class of curves included in the equation
(x/a)n + (y/b)n = 1;
we shall now consider the equation
rn -= an cos na,
which includes many important and well known curves.
By the ordinary formula
tan q = rdO/dr = - cot n0,
whence
- = - + n0.
Accordingly if (p, X) be the coordinates of Y,
x = + - 7r= ( + 1)0,
p = r sin ( = r cos n0.......................(20),
whence
pf+l = an+l cos x/(?z + 1).................. (21).
Equation (21) is the pedal of the curve; from which it follows
that every pedal is a curve of the same species, and that each successive pedal is obtained from the preceding one by changing n
into n/(n + 1). The reciprocal polar is the curve
n 12
cn+1 = rn+l cos nx/(n + 1)............... (22),
and is obtained by changing n into - n/(n + 1).
From (20) we obtain
a lp r...........................(23),
which is the p and r equation of the curve. The radius of curvature is
dr? ali
P = p (n =....... (24)...................( 2~).
Orthoptic Loci.
68. The orthoptic locus of a curve is the locus of the point of
intersection of two tangents which cut one another at right
angles. If the two tangents are inclined at a constant angle, the
locus is called the isoptic locus.