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

THE EVOLUTE OF AN ELLIPSE.

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The Evolute of an Ellipse.
336. Let (x, y) be the coordinates of the centre of curvature
of a point on an ellipse whose excentric angle is b; and let sr be
the angle which the normal makes with the major axis. Then
x = a cos ( - p cos 'r,
and a cos * =p cos o, pp = a2 sin2 + + b2 cos2 9,
whence ax = (a2- b2) cos3 (,
by = (a2 - b2) sins 3.
Accordingly the equation of the evolute is
(ax) + (by) = (a2- b)
The form of this equation shows that we cannot deduce
properties of the four-cusped hypocycloid by supposing an ellipse
to degrade into a circle; but if the equation be written in the
form
(x/A)2 + (y/B) =,
properties of the evolute may be deduced from those of the
hypocycloid by orthogonal projection.
337. In the figure, let CZ be the perpendicular from the
centre of an ellipse on to the normal at P, 0 the centre of curvao
ture, and OQ perpendicular to PO. Then if CZ=p', ZCA = -Tr-*,
it follows from the properties of the ellipse that
_ (a2- b2) sin ' cos A
p Z (a2 cs2k + b2 sin2 #)t'
and therefore the pedal of the evolute is the sextic curve
(a2y2 + b2x2) (X2 + y2)2 = (a2 - b2)2 x2y2
whilst the orthoptic locus is the sextic curve given at the end of
~ 68.