When is torsion zero

Can someone please explain this proof to me. I know that a circle is planar and has nonzero constant curvature, so this must be an exception, but I am a little lost on the proof.

A circle is indeed planar, and has constant nonzero curvature, but the torsion of a circle is zero; it's not an exception. This being the case, we may write. It is easy to see that. Careful scrutiny of 25 suggests that.

Thus we have. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.

Create a free Team What is Teams? Learn more. A space curve is planar if and only if its torsion is everywhere 0 Ask Question. Asked 7 years, 1 month ago. Active 2 months ago. The best answers are voted up and rise to the top.

Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Conditions that torsion is zero in a space curve Ask Question. Asked 11 years ago. Active 11 years ago.

Viewed 7k times. Do you want a condition for torsion to be zero at a particular point but not necessarily elsewhere? Any ideas or it is just not possible?

Add a comment. Active Oldest Votes. It stands to reason that zero torsion gets you a flat curve, as removing the appropriate terms in the Frenet-Serret formulae show. Jack Lee Jack Lee I had completely overlooked that. So I guess my example is, at best, a curve with torsion that's identically zero where it's defined, which is everywhere but the origin.

Is it possible to define torsion when the curvature vanish? I guess do Carmo means that one can extend the torsion from that set to the whole curve continuously. Note that when the torsion in some point is nonzero, the curve "goes out" of its osculating plane in that point and, therefore, a tridimensional curve, defined in an interval, is planar if and only if its torsion is zero note that in the latter fact we are assuming that torsion is everywhere defined in the interval and that only happens if curvature is always greater than 0.

For full functionality of this site JavaScript must be enabled. Follow these instructions on how to enable JavaScript in your web browser and reload the page.