286. DEF. The locus of the centres of curvature of all points of a given plane curve is called the evolute of that curve. If the evolute itself be regarded as the original curve, a curve of which it is the evolute is called an involute.

The equation of the evolute of a given curve may be found by eliminating x and y between equations (a), (B) of the last article and the equation of the curve.

Ex. To find the locus of the centres of curvature of the parabola

287. Evolute touched by the Normals.

2'

3

2

3

Let P1, P, P, be contiguous points on a given curve, and let the normals at P,, P, and at P,, P, intersect at Q1, Q2 respectively. Then in the limit when P„, P, move along the curve to ultimate coincidence with P, the limiting positions of Q1, Q2 are the centres of curvature corresponding to the points P1, P, of the curve. Now Q and Q, both lie on the normal at P,, and therefore it

2

is clear that the normal is a tangent to the locus of such points as Q1, Q2, i.e., each of the normals of the original curve is a tangent to the evolute; and it will be seen in the chapter on Envelopes (Art. 313) that in general the best method of investigating the equation of the evolute of any proposed curve is to consider it as the envelope of the normals of that curve.

288. There is but one Evolute, but an infinite number of Involutes.

Let ABCD... be the original curve on which the successive points A, B, C, D, ... are indefinitely close to each other. Let a, b, c, be the successive points of

intersection of normals at A, B, C, ... and therefore the centres of curvature of those points. Then looking at ABC... as the original curve, abcd... is its evolute. And

regarding abcd... as the original curve, ABCD... is an

If we suppose any equal lengths AA', BB′, CC′, ... to be taken along each normal, as shown in the figure, then a new curve is formed, viz., A'B'C'..., which may be called a parallel to the original curve, having the same normals as the original curve and therefore having the same evolute. It is therefore clear that if any curve be given it can have but one evolute, but an infinite number of curves may have the same evolute, and therefore any curve may have an infinite number of involutes. The involutes of a given curve thus form a system of parallel

curves.

289. Involutes traced out by the several points of a string unwound from a curve.

Since a is the centre of the circle of curvature for the

point A (Fig. 57), aA=aB

aA-gG arc ab+arc bc+...+arc fg

=

=arc ag.

Hence the difference between the radii of curvature at two points of a curve is equal to the length of the corresponding arc of the evolute. Also, if the evolute abc... be regarded as a rigid curve and a string be unwound from it, being kept tight, then the points of the unwinding string describe a system of parallel curves, each of which is an involute of the curve abcd..., one of them coinciding with the original curve ABC.... It is from this property that the names involute and evolute are derived.

290. Radius of Curvature of the Evolute.

It is easy to find an expression for the radius of curvature at that point of the evolute which corresponds to any given point of the original curve.

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b

Let 0 (Fig. 57) be the centre of curvature for the point a of the evolute. The angle dy between the normals at = the angle between the tangents at a, b

= the angle between the tangents at A, B to the original curve

And if s' be the arc of the evolute measured from some fixed point up to a, and p' the radius of curvature of the evolute at ɑ, and ρ that of the original curve at A, we have, rejecting infinitesimals of order higher than the first, ds' = arc ab= dp,

s being the arc of the original curve measured from some fixed point up to A, and the angle which the tangent at A makes with some fixed straight line.

INTRINSIC EQUATION.

291. The relation between the length of the arc (s) of a given curve, measured from a given fixed point on the curve, and the angle between the tangents at its extremities () has been aptly styled by Dr. Whewell the Intrinsic Equation of the curve. For many curves this relation takes a very elegant form. The name seems specially suitable to a relation between such quantities as these, depending as it does upon no external system of co-ordinates. The method of obtaining the intrinsic equation from the Cartesian or polar relation is dependent in general upon processes of integration. If the equation of the curve be given as y=f(x), the axis of x being supposed a tangent at the origin, and the length of the arc being measured from the origin, we have

If s be determined by integration from (2) and x elimin