Dr. Robert Lopez

Evolute of a Space Curve

Given a plane curve r(t) , its evolute R(t) is another plane curve defined in one of two ways, either as the locus of the centers of curvature, or as the envelope of the normal lines along r. The curve also generates an infinite family of involutes, curves whose tangents are orthogonal to r. Thus, r is one member of the family of involutes of R.

The two definitions of the evolute of a plane curve are easy to implement, visualize, and apply. In fact, a previous webinar "A Tale of Two Involutes" explores the evolute and involutes of an ellipse, and shows that the involutes of the evolute form a family of curves parallel to the ellipse.

The evolute of a space curve is not the locus of any center of curvature, neither the centers of the circles of curvature, nor the centers of the spheres of curvature. Instead, the evolute of a space curve is another space curve whose tangents are orthogonal to the given curve. This is a generalization of the second definition of the evolute of a plane curve, a generalization that, unlike in the plane case, implies there are an infinite number of evolutes for a space curve.

This webinar will provide a quick review of the plane case, then derive formulas for the evolute of a space curve. An extended example will concentrate on finding the evolute(s) of a helix, and visualizing this infinite family as a surface in space. It will give analytic and visual verification that tangents along an evolute are indeed orthogonal to the helix, and even that the principal normals along the evolute are parallel to the tangents on the helix.
Language: English
Duration: 50 Minutes
Related Terms: Evolute, Spacecurve, Involute