Timeline for Space curves and torsion
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 3, 2014 at 5:17 | vote | accept | Chaitanya | ||
| Dec 27, 2013 at 17:59 | comment | added | Robert Bryant | Actually, your last statement is not true at all. For example, if you take $\kappa$ and $\tau$ to be constant (and nonzero), which is about as periodic as you can get, the resulting space curve (a helix) will certainly not be periodic. In fact, determining when a pair of periodic functions is the curvature and torsion of a periodic space curve is highly nontrivial. | |
| Dec 27, 2013 at 15:48 | comment | added | Manfred Weis | @Robert: Ok, the Archimedean spiral is not a closed curve, but that can be easily fixed by using instead a curve in the plane, which is the hull curve of a convolution of an Archimedean spiral with "very small" circles, whose center is on the spiral, letting the parameter $t$ of the planar curve tend to infinity. Your other question is more delicate but it can be answered, that the integration of periodic functions for curvature and torsion essentially results in periodic functions for the three coordinates of the space curve as a consequence of the Frenet-Serret formulas. | |
| Dec 27, 2013 at 13:44 | comment | added | Robert Bryant | You appear to be ignoring the OP's condition that the curve be closed, which, of course, rules out curvature-torsion profiles such as Archimedean spirals, etc. Also, without producing an example where it doesn't, how do you know that the curvature-torsion profile curve of a closed space curve won't always touch at least one corner of its bounding box? I agree that it looks very unlikely on the face of it, but that, in and of itself, is not evidence for belief. | |
| Dec 27, 2013 at 10:47 | history | answered | Manfred Weis | CC BY-SA 3.0 |