In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. Equivalently, it is the envelope of the normals to a curve. The original curve is an involute of its evolute. (Compare and )
and the unit normal to the curve is the unit vector N(s) perpendicular to T(s) chosen so that the pair (T,N) is positively oriented.
The curvature k of γ is defined by means of the equation
for each s in the domain of γ. The radius of curvature is the reciprocal of curvature:
The radius of curvature at γ(s) is, in magnitude, the radius of the circle which forms the best approximation of the curve to second order at the point: that is, it is the radius of the circle making second order contact with the curve, the osculating circle. The sign of the radius of curvature indicates the direction in which the osculating circle moves if it is parameterized in the same direction as the curve at the point of contact: it is positive if the circle moves in a counterclockwise sense, and negative otherwise.
The center of curvature is the center of the osculating circle. It lies on the normal line through γ(s) at a distance of R from γ(s), in the direction determined by the sign of k. In symbols, the center of curvature lies at the point:
As s varies, the center of curvature defined by this equation traces out a plane curve, the evolute of γ.
where the unit normal N = (−sinφ, cosφ) is obtained by rotating the unit tangent T = (cosφ, sinφ) through an angle of 90°.
The equation of the evolute may also be written entirely in terms of x, y and their derivatives. Since
R and φ can be eliminated to obtain:
This follows by differentiation of the formula
and employing the Frenet identity N′(s) = −k(s)T(s):
from which it follows that dσ/ds = |dR/ds|, as claimed.Unit tangent vector Another consequence of is that the tangent vector to the evolute E at E(s) is normal to the curve γ at γ(s).Curvature The curvature of the evolute E is obtained by differentiating E twice with respect to its arclength parameter σ. Since dσ/ds = |dR/ds|, it follows from that
where the sign is that of dR/ds. Differentiating a second time, and using the Frenet equation N′(s) = −k(s)T(s) gives
As a consequence, the curvature of E is
where R is the (signed) radius of curvature and the prime denotes the derivative with respect to s.Relation with involuteIntrinsic equation If φ can be expressed as a function of R, say φ = g(R), then the Whewell equation for the evolute is Φ = g(R) + π/2, where Φ is the tangential angle of the evolute and we take R as arclength along the evolute. From this we can derive the Cesàro equation as Κ = g′(R), where Κ is the curvature of the evolute.
By the above discussion, the derivative of (X, Y) vanishes when dR/ds = 0, so the evolute will have a cusp when the curve has a vertex, that is when the curvature has a local maximum or minimum. At a point of inflection of the original curve the radius of curvature becomes infinite and so (X, Y) will become infinite, often this will result in the evolute having an asymptote. Similarly, when the original curve has a cusp where the radius of curvature is 0 then the evolute will touch the original curve.
This can be seen in the figure to the right, the blue curve is the evolute of all the other curves. The cusp in the blue curve corresponds to a vertex in the other curves. The cusps in the green curve are on the evolute. Curves with the same evolute are parallel.
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