1.6.4 Torsion
Curvature was a scalar function measuring the "curviness" of a curve at various points. We defined curvature to be an inherent property of a curve, meaning it did not depend on parameterization. We did this by using an arclength parameterization to define (though we derived formulas that allowed us to avoid the tedious step of actually finding an arclength parameterization).
For space curves we have a second scalar measure - torsion. Torsion measures how rapidly a curve is twisting away from its osculating plane. Low torsion indicates the curve is nearly planar and high torsion indicates the curve is twisting away from its osculating plane very quickly.
Defining torsion requires first making a series of deductions about the vectors in the Frenet frame. These are called the Frenet-Serrett formulas.
If is a twice-differentiable, regular, arclength parameterization of a curve, then
The constant referenced in the Frenet-Serret formulas is the torsion of the curve.
Again we can derive a formula for torsion that avoids the tedium of finding an arclength parameterization:
The GeoGebra applet below displays a graph of torsion alongside the corresponding curve.
Show that a curve lying entirely in the plane has zero torsion.