# Characterization of convex space curve

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.

• Any set $A$ lies entirely on its convex hull by definition: $A\subset CH(A)$. Or did I missunderstood something in your question? – Luc Jul 30 '13 at 9:22
• I think it means "on the boundary of its convex hull". – alvarezpaiva Jul 30 '13 at 9:40
• 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 – Sunayana Ghosh Jul 30 '13 at 9:42
• 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? – Joseph O'Rourke Jul 30 '13 at 11:53
• 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. – alvarezpaiva Jul 30 '13 at 12:38