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

EPICYCLOIDS AND IYPOCYCLOIDS.

211

and four-cusped hypocycloids possess many remarkable properties,
and will be discussed separately.
318. In the preceding section we have tacitly supposed that
the radius of the fixed circle is greater than that of the moving
circle; we shall now show that if the radius of the latter is greater
than that of the former, the hypocycloid becomes an epicycloid
generated by a rolling circle whose radius is equal to the difference
between the radii of the two circles.
Let AQB and QPR be the fixed and rolling circles, 0, 01 their
centres; a and b their radii; and let E be the point with which
C
B E
2OR R
0 0
A
P was initially in contact. Let
EOQ = 0, QOP =.
Let R be the other extremity of the diameter through Q of
the moving circle, and draw RP to meet the diameter AB of the
fixed circle in C.
Since BQA and CPA are right angles, and OQ = OB, it follows
that CB = QR; whence
AC = 2a + BC = 2 (a + b);
whence C is a fixed point, and a semicircle can be described
through APC whose radius is a + b, and centre 02.
Again are AE= a (0 + = (a + b),
arc AP = (a + b) A02P = (a + b),
whence arc AP = arc AE. Accordingly the epicycloid which is
the locus of P may be generated by the circle APG whose radius
is a+ b rolling on the fixed circle in such a manner that their
concavities are in the same direction.

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