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

222 MISCELLANEOUS CURVES.
draw OEQ so that EQ= OE. Draw O Y, QP perpendicular to
AB; let AOY= b, (x, y) the coordinates of P.
Since QA, Q B are perpendicular to OA, OB; PB is the
direction of motion of P, and therefore AB touches the curve
enveloped by it at P. Now
x = BP sin 0 = A Ysin b
= a sin3 l,
similarly y = a cos3 q,
whence the locus of P is the hypocycloid
x' + y3 = at.
Accordingly the four-cusped hypocycloid is the envelope of a
straight line of constant length, which slides between two straight
lines at right angles to one another.
The equation of the pedal and the p and r equation are
respectively

r = ~a sin 20,
r2 = a2 - 3p2,
p = - rdr/dp = 3p,
n is

whence
and the intrinsic equatioi

s = a sin2 r,
where = r.
334. The orthoptic locus is a curve similar to the pedal.
Let AB and CD be perpendicular tangents intersecting at T;
let OT= r, TOA = 0. Then since CD = a, OCD = b,
TY=acos sin = OY,
whence 0 = r +, r = p2,
accordingly r = - (a/21) cos 20.
335. If the tangent at P is the normal at T, then OP = OT.
From the p and r equation we have
OT2 = a2- 3TY2 = a2 - 3p2,
whence OT= OP.