Reconstruction from Curvature and Torsion

Given positive functions s(t) and k(t) and a function t(t) , can we always find a curve X(t) such that the arclength of X(t) is s(t), the curvature is k(t) and the torsion is t(t)? It turns out that we can always find a unique such curve defined over an interval a<t<b , once we impose the initial conditions X(a) =0 , T(a) =(1,0,0) , and P(a) =(0,1,0) . The proof of this theorem is much more complicated than the proof of the corresponding theorem in the plane. However, it is still possible to construct an approximation to the curve by using a stepwise process similar to the one used to reconstruct a plane curve given its speed, its curvature and its initial position and the direction of its tangent vector at the initial point.

Need to put an APPLET here! Reconstruction of a Space Curve from Curvature and Torsion Need to put an IMG here!

To see the curve you must turn on the Draw Curve checkbox. In order to make a series of changes in the control panel more quickly, turn the checkbox off first. You should also keep the resolution under 50 for a quick response.

This demo shows the reconstruction of a space curve from arbitrary speed , curvature, and torsion functions. For the reconstruction, the initial point of the curve will always be at the origin and the initial tangent vector will be T=(1,0,0) . This identifies a single curve in the family of solutions to the simultaneous differential equations involved in reconstructing a space curve from curvature and torsion.

The basic theorem on helices predicts that if the torsion is a constant multiple of the curvature, then there will be a direction which makes a constant angle with all tangent directions. Identify this direction for various choices of the curvature function and the constant ratio of curvature and torsion.