Involute Curves of Space Curves

In the study of plane curves, once we defined the evolute of a curve, we considered the inverse problem of finding all curves that have a given curves as evolute. We may consider the analogous problem for space curves. If a curve Y is the evolute of a curve X, then X is said to be an "involute" of the curve Y. As in the planar case, we may define an involute by

In this way, for various choices of the constant c, we obtain a one-paramater family of space curves that are involutes of a given curve.

Need to put an APPLET here! Involutes of Space Curves Demo Need to put an IMG here!

Clicking on Involute will toggle the display of the involute

As in previous demos, the variable c is a constant that is to be used in the type-in curve while a is the constant that is used in the definition of the involute.

This simple demo allows you to input a curve with its domain and will show its involute.

We look at the involute of a helix.

For a circular helix, the involute will be a plane curve, which is itself an involute of a circular cross-section of the cylinder. The same will be true for any helix drawn on a cylinder over a plane curve.

A helix is defined to be a curve for which the unit tangent vectors make a fixed angle with a given unit vector W . If

with a being constant, then so Since W is a unit vector, z=±sin(a) so that Differentiating this relationship gives so Thus we have a linear relationship between t(t) and k(t) independent of t . Conversely if we have for all t , then if we may define the vector cos(a)T(t)±sin( a)B( t) and differentiate to show that it must be a constant vector, say W . This W will make a constant angle with all unit tangent vectors.

To show that an involute of a helix with axis vector W is a plane curve, we observe that

so so and the condition for the involute to lie in a plane perpendicular to W .