Curvature-Torsion Defined Curve

Given torsion and curvature functions, compute the curve uniquely.

Description

Given continuous curvature and torsion as functions of t, a unique 3D curve is identified by solving a system of 3 vector differential equations, known as Frenet's equations. Each equation refer to one the vector basis: tangent, normal and binormal vectors. Each equation can be split into 3 scalar equation whose unknowns are the coordinates of each vector.

The resulting system is linear with a 9x9 antisymmetric matrix. It is solved by means of ode45 solver.

• MATLAB release: MATLAB 7.0.1 (R14SP1)

[Kartik Venkataraman]

Prof. Paolo Panarese,
I downloaded the software and tried it out. It is very useful to visualize the relationships between curvature, torsion, and form. I do have one question though. If instead of a closed form parameterization of the curvature, I wanted to define it as actual values, how could I modify the code to generate the curve?

In other words, for example, instead of saying that the curvature was (t^2), I would like to state the actual curvature at each point of the curve. So if the curve was defined by 10 points, I would have the actual curvature at each of those 10 points.

How can I modify the code to enable me to generate this?

I apologize if this seems trivial. But I am kind of new to solving ODE's with Matlab, and I could definitely use some helpful suggestions.

Regards,
Kartik Venkataraman
University of California
Santa Cruz

[Ken Crandall]

A curve with constant curvature of 10 and constant torsion of 10 is a straight right handed helix. That is not what I get when running the program.

[Ben Friedrich]

Dear Ken,

carefully check the different scalings of the axes. Try kappa=3, tau=4, T=[3/5 0 4/5],
N=[0 1 0] for a helix pointing in z-direction.

Ben