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

212

MISCELLANEOUS CURVES.

Trochoidal Ctrves.
319. The trochoid is a curve described by any point in the
plane of a circle which rolls on a straight line.
Trochoids are sometimes called prolate cycloids and curtate
cycloids according as the point lies within or without the rolling
circle. The equations of a trochoid can easily be shown to be
x= a(O- m sin 0)(19)
y = a ( - m cos 0)
where m is less or greater than unity according as the point lies
within or without the circle.
320. When a circle rolls on another circle, the locus of any
point in the plane of the moving circle is called an epitrochoid or
a hypotrochoid according as the latter rolls on the exterior or the
interior of the fixed circle.
If c be the distance of the point from the centre of the rolling
circle, the equations of the epitrochoid are
= (a + b) cos - os (a + b) 0/b. 20
y = (a + b) sin 0-c sin (a + b) 0/b ( )
whilst those of the hypotrochoid are obtained by changing the
sign of b.
When a = b, the epitrochoid becomes a limagon.
Transfer the origin to the point x = - c; then (20) become
x = 2 (a - c cos 0) cos 0,
y = 2 (a - c cos 0) sin 0,
whence 0 is the vectorial angle, and the polar equation is
r = 2 (a - c cos 0),
which is a limagon.
In (20), put
a = (m - )b, mb = c..................(21)
and change the direction of the axis of y, and we obtain
x = 2c sin ~(m + 1)0 sin (n(m - 1) 0,
y = 2c cos I(m + 1) 0 sin I (m - 1) 0;
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