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

210

MISCELLANEOUS CURVES.

316. To find the tangential equation of an epicycloid.
Through 0 draw a line OY' perpendicular to OT cutting the
tangent in Y'; then
OT sin 2
OE sin ( + 0)'
whence s 2a/bl)0
en e (a + 2b) sin a/b..................(12)
O Y' sin I b
OE cos ( +0)'
w cos (Qa/b + 1) 0
whence (13)
v (a + 2b) sin aOl/b.....
the elimination of 0 between these equations furnishes the required result.
The one-cusped epicycloid, as we have already shown, is the
cardioid, and its tangential equation is
27a2 (2 + 2)(1 - a ) = 4...............(14),
whilst those of the two- and four-cusped epicycloids are
4a2 (: + 2)(l - at )= 1..................(15)
and (6b)6 (,2 + q2) 2y2 = 4 {27b2 (~2 + q2) - 1}2........ (16)
respectively. The former curve is of the fourth class and it
cannot be of lower degree than the sixth, since the common
tangent at the two cusps has a contact of the second order with
the curve at these points. Its Cartesian equation is
4 (x2 + y2 - a2)3 = 27a4y2.................. (17).
317. The hypocycloid is the curve traced out by a point on
the circumference of a circle which rolls inside a fixed circle.
The coordinates of any point on a hypocycloid can be shown
to be
x = (a - b) cos 0 + b cos (a - b) 0/b (18).
y = (a-b) sin 0-bsin(a - b) 0/bJ ("
whence the corresponding results for a hypocycloid can be deduced
from those of an epicycloid by changing the sign of b.
A two-cusped hypocycloid is a diameter of the fixed circle,
as can be at once shown by elementary geometry. The three