4.3: 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.

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.

Investigate the curves that result from a constant function k(t)=1 R , a constant function t(t)=w , and a constant speed s^'(t)=1 .

Investigate the curves that result from the functions k(t)=t , t(t)=t , and s^'(t)=1 . (This means the value of k(t) and t(t) are equal to t for all t .)

Investigate the curves that result when we enter the curvature and torsion of a known curve as our k(t) and t(t) . To simplify this, erase what is in the Curvature type-in and enter Curvature Helix if you want the curvature of the helix. Enter Speed Helix and Torsion Helix in the ds/dt and Torsion type-ins.

You can also enter the curvature, speed, and torsion of arbitrarily defined functions. For example, if you want to see the reconstruction of X(t) =(e^tcos(t) ,e^tsin(t) ,t) , you can enter Curvature(R->R^3:t->[exp(t)*cos(t),exp(t)*sin(t),t]) , and so on.

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.

When there is a linear relation between the curvature and the torsion, so that the torsion is a constant multiple of the curvature plus another constant, then the curve is called a "Bertrand curve". In such a case, there is another Bertrand curve, called a "Bertrand mate" such that the segment joining corresponding points is perpendicular to the tangent line for each of the curves. Every helix is a Bertrand curve. How many mates will such a curve have? Investigate some other Bertrand curves. Are there any closed Bertrand curves?