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

208

MISCELLANEOUS CURVES.

This is the tangential polar equation of an epicycloid and is of
the form p = c sin nO; it is also the pedal of the curve with respect
to the centre of the fixed circle, and the inverse curve is the
reciprocal polar of the epicycloid.
E
C 0'
0` T X
d2p 4b (a 4-+ b)
Again P=P+d2- (a2b)..................(8),
which shows that the radius of curvature is proportional to the
perpendicular from the centre of the fixed circle on to the tangent.
Also since p = ds/dr, we obtain from (7) and (8)
s -4b(a+b)1 - cos...............(9),
a a + 2b..(9)
which is the intrinsic equation of the curve, s being measured
from A.
The p and r equation of the curve seems to have been first
given by the Jesuits in their notes to Prop. LI. of Newton's
Principia, and may be obtained as follows. Let OP=r; then
r2= (a + b)2 + b2 - 2 (a + b) b cos
=a2 +4 (a+b)b sin2 ~q
4 (a + b) bp2
-a ' (a + 2b)2
(a + 2b)2
whence p2= (a + 2b)(
whence 2 = + (r2- a2)........................(10).
4 (a + b) b
313. The evolute of an epicycloid is a similar epicycloid.
The evolute of the curve is the envelope of the normal PQ.
Now if OZ be the perpendicular from 0 on to the normal
OZ= a cos 2 p = a cos a*r/(a + 2b),

which is the tangential polar equation of a similar epicycloid.