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8
votes
2answers
335 views

Surface in 3D that realizes all pairs of principal curvatures

This is a question that Willie Wong raised in comments after he answered my question, Surface analog of clothoid: curvatures covering $\mathbb{R}^3$. Willie's question is more interesting (and ...
12
votes
2answers
487 views

On closed simple curve with curvature at most 1

I am looking for the reference to the following theorem. I have to apply a similar statement, and it would be nice to trace the source. Please note, I know few proofs in fact it is Problem 3 in my ...
8
votes
1answer
490 views

Geodesics on the twisted pseudosphere (Dini's surface)

I wonder how difficult it is to compute geodesics on Dini's Surface, a twisted pseudosphere? Here is one parametrization, from Alfred Gray's Modern Differential Geometry of Curves and Surfaces, ...
5
votes
1answer
144 views

Curvature flows for PL closed curves in the plane?

I'm curious to what extent people have studied "curvature flows" on PL closed curves in the plane. There's a paper by Gage and Hamilton from 1986 that describes the long-term behaviour of smooth ...
3
votes
2answers
430 views

Surfaces in $\mathbb R^3$ with negative curvature bounded away from zero

Is there a surface in $\mathbb R^3$ which is a closed subset and whose curvature is negative and bounded away from zero? And the small-print... By surface I mean smooth surface without ...
1
vote
3answers
260 views

Surface analog of clothoid: curvatures covering $\mathbb{R}$

The clothoid $C$, a.k.a. the Euler spiral, is one among many curves with the property that its curvatures cover $\mathbb{R}$ in the sense that, for every $x \in \mathbb{R}$, there is a point $p \in C$ ...