How can i understand the graphical interpretation of Torsion of a curve?

Question

I understand the graphical interpretation of the curvature of a curve in $\mathbb{R}^3$. Could you help me to understand the graphical meaning of the torsion of a curve? I know that if torsion is positive, the curve goes through the osculating plane from below upwards. Conversely, if torsion is negative, the curve goes through the osculating plane from above downwards.

Question: How can I interpret torsion geometrically? For example, the curvature is the inverse of the radius of the osculating circle. Is there a similar interpretation for torsion?

Answer 1

Here's how my differential geometry professor explained it to me.

• Curvature measures the failure of a curve to be a line. If $\gamma$ has zero curvature, it is a line. High curvature (positive or negative corresponding to right or left) means that the curve fails to be a line quite badly, owing to the existence of sharp turns.
• Torsion measures the failure of a curve to be planar. If $\gamma$ has zero torsion, it lies in a plane. High torsion (positive or negative corresponding to up and down) means that the curve fails to be planar quite badly, owing to it curving in various directions and through many planes.

Now for some examples.

• $\tau = 0, \kappa = 0$: A line. Lines look very much like lines, and they are certainly planar.
• $\tau = 0, \kappa =k > 0$: A circle. Circles don't look like lines, especially small ones. They have constant curvature. However, they do lie in a plane.
• $\tau = c >0, \kappa = k >0$: A helix. Helixes curve like circles, failing to be lines. They also swirl upwards with constant torsion, failing to lie in a plane.
• $\tau >0 , \kappa = k > 0$: A broken slinky. Slinkies curve like circles, failing to be lines. They generally have constant positive torsion, like helixes. But if you break them, the torsion remains positive (viewed from the bottom up), but how large the torsion is corresponds to how stretched the slinky is. A very stretched slinky has large torsion, compacted slinkies have small torsion.

Answer 2

You say you understand curvature, so let's work from there.

The curvature at a given point $P$ is a measure of how fast the curve moves away from its tangent line at $P$. Saying it another way, it measures the amount on non-linearity at $P$.

Torsion is very similar: The torsion at a given point $P$ is a measure of how fast the curve moves away from its osculating plane at $P$. In other words, it measures the amount on non-planarity at $P$.

Answer 3

Torsion is the speed with which the osculating plane rotates around the curve in radians per unit length. Positive torsion indicates a clockwise rotation and negative torsion a counter clockwise rotation. Torsion is an infinitesimal notion, so informally it measures speed of rotation on small portions of the curve where it is an almost straight line. It is hard to see this rotation globally since the curve itself also twists through space.

Imagine a striped garden hose that follows a closed space curve. The hose ends meet with a certain rotation, which you can see by tracing the stripes along the hose. This angle is the integral of the torsion along the curve, its total torsion.