Evolute Totally Explained

Everything about Evolute totally explained

In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. Equivalently, it's the envelope of the normals to a curve. The original curve is an involute of its evolute. (Compare and )

History

Apollonius (c. 200 BC) discussed evolutes in Book V of his Conics. However, Huygens is sometimes credited with being the first to study them (1673).

Equations

Let

(x, y) = (x(t), y(t))

be a parametrically defined plane curve. Let

R = 1/kappa

be the radius of curvature and

phi

be the tangential angle. Then the center of curvature at

(x, y)

is given by

(x - R sin phi, y + R cos phi)

and we may take

(X, Y) = (x - R sin phi, y + R cos phi)

as parametric equations for the evolute. We have

(cos phi, sin phi) = frac)

Examples

The evolute of a parabola is a semicubical parabola. The cusp of the latter curve is the center of curvature of the parabola at its vertex.

The evolute of a Logarithmic spiral is a congruent spiral.

The evolute of a cycloid is a similar cycloid.Further Information

Get more info on 'Evolute'.

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