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

THE THREE-CUSPED HYPOCYCLOID.

217

326. To find the tangential polar and the intrinsic equations of
the curve.
Let OY=p, OE=a, OtY= 0,
then since we have shown that OEY= 30, it follows that
p = a sin 30,
or p = - a cos 30,
if 4 = 1r - 0. The intrinsic equation is
s= a(1 -cos 30),
and the p and r equation is
r = 8p- 9a2.
327. The portion of the tangent contained within the curve is
of constant length.
Join PP' and draw OZ perpendicular to PP'. Since O'PE
and OET are a pair of equal isosceles triangles, PE= ET;
similarly TF= FP', and therefore EF is parallel to PP'. Whence
PP'= 2EF= 40E.
Produce TO to meet PP' in H; then since 0 is the middle
point of TH, it follows that OZ= OY; hence PP' touches the
hypocycloid.
The line OH bisects PP' in H; whence if two tangents be
drawn to a three-cusped hypocycloid which are at right angles to
one another, the chord of contact is also a tangent to the curve,
and is bisected by the line joining the centre 0 with the point of
intersection of the perpendicular tangents.
328. If three tangents be drawn to a three-cusped hypocycloid,
two of which are at right angles, the third tangent is perpendicular
to the chord of contact of the other two.
Draw TZ' perpendicular to PP' cutting OE in Z'; let
EOA =, EtO =0.
Then since OZ'T is a right angle
OZ' = OTcos TOZ'.
But OT= a, and TOZ'= r - 2TEO = 7- 60 = 7r - 30;
whence OZ' = - a cos 30,
and therefore TZ' touches the hypocycloid.