2
$\begingroup$

In general, a smooth curve $C$ in $R^3$ that lies entirely on the boundary of its convex hull, $\mathcal{H}(C)$, is defined to be convex.

  1. Does any one know of a characterization of a curve in space as convex depending on the sign of its torsion $\tau$?
  2. Is the projection of a non-singular "short" curve in space to any plane always convex?

Where a "short" curve in space is such that there is a 1-1 correspondence between the space curve and the corresponding projected curve in a plane.

Thanks.

Improve this question
$\endgroup$
8
  • 1
    $\begingroup$ Any set $A$ lies entirely on its convex hull by definition: $A\subset CH(A)$. Or did I missunderstood something in your question? $\endgroup$
    – Luc
    Jul 30, 2013 at 9:22
  • 1
    $\begingroup$ I think it means "on the boundary of its convex hull". $\endgroup$
    – alvarezpaiva
    Jul 30, 2013 at 9:40
  • $\begingroup$ Sorry about not being precise. I am referring to the definition of a convex curve as provided by Sedykh in link.springer.com/article/10.1007%2FBF01077070#page-1 $\endgroup$
    – Sunayana Ghosh
    Jul 30, 2013 at 9:42
  • $\begingroup$ I cannot make sense of the 2nd question. Perhaps this is what is meant: If some curve $C$ in space has the property that every projection of $C$ to a plane is simple, i.e., has no self-intersections, must then $C$ lie on its convex hull in 3D? $\endgroup$
    – Joseph O'Rourke
    Jul 30, 2013 at 11:53
  • 4
    $\begingroup$ In any case it is impossible that all projections be simple: join two points of the curve by a segment and project in the direction perpendicular to it. The most you can ask is that almost all projections be simple. I guess this only says the curve is planar. $\endgroup$
    – alvarezpaiva
    Jul 30, 2013 at 12:38

1 Answer 1

Reset to default
2
$\begingroup$

The first question is not tractable, as one can glean by reading this paper by Bates and Melko (whence one learns that it took a century to produce a curve of constant torsion and non-vanishing curvature.) However, this paper has a lot of interesting stuff if you are interested in this circle of questions.

Improve this answer
$\endgroup$
2
  • $\begingroup$ That's an interesting reference! I like their CONJECTURE 3.3. There are no closed curves of nonzero constant torsion on an ovaloid. $\endgroup$
    – alvarezpaiva
    Jul 30, 2013 at 17:19
  • $\begingroup$ Thanks for the link to the article. The notion of signed curvature for space curves seems interesting. $\endgroup$
    – Sunayana Ghosh
    Aug 2, 2013 at 11:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.