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

220

MISCELLANEOUS CURVES.

Draw NY perpendicular to OL, and let NY=p, OLE= f;
then p = R cos ON Y,
and ONY= 1- NOE + =OFE J+;
also since FOE = A,
OFE= r-A -OEF
= C-A + 2f,
whence ON Y= 7- A + 3b,
and therefore p = 4 R cos (30 + C - A).
If a triangle be self-conjugate to a rectangular hyperbola, it is
known that the centre of the hyperbola lies on the circle circumscribing the triangle, and that the curve passes through the
centres of the inscribed and the three escribed circles of the
triangle. Hence the preceding theorem shows that the envelope
of the asymptotes of all rectangular hyperbolas to which a given
triangle is self-conjugate is a three-cusped hypocycloid, whose
centre is the centre of the circumscribing circle of the triangle.
The Four-cusped Hypocycloid.
332. Putting a= 4b in (18) of ~ 317, the equations of the
curve become
x = 3b cos 0 + b cos 30= a cos3 0
y = 3b sin 0 - b sin 30= a sin3 0f
whence the Cartesian equation is of the form
3 + y = a3...........................(2).
The curve therefore belongs to the class of curves discussed in
~ 55. Equation (1) may also be expressed in the form
(a2 - _ - y2)3= 27a2x22..................(3),
whilst its reciprocal polar is
1 1 1
I +............................ (4),
X2 y2 '2
which shows that the hypocycloid is of the sixth degree and fourth
class.
The characteristics of the curve are most easily investigated
by means of the reciprocal curve.