All Questions
16 questions
0
votes
0
answers
34
views
Converse of Scherk–Segre theorem on the number of vertices of a convex space curve
It is well known that any smooth simple closed convex curve $\gamma$ in $\mathbb{R}^{3}$ that meets no plane in more than 4 points has exactly 4 vertices, i.e., points of vanishing torsion; here "...
- 965
4
votes
1
answer
139
views
Characterization of convexity by connectedness of hyperplane sections
Let $S$ be a smooth closed connected embedded hypersurface in $\mathbb R^n$.
Is it true that $S$ is convex, i.e. is a boundary of a convex set, if and only if any section of $S$ by a hyperplane is ...
- 2,934
6
votes
1
answer
444
views
Can every smooth space curve be realized as an origami curved crease?
Many years ago, Ron Resch told me that he proved that every smooth simple space curve $C$ could be realized as a curved crease $\gamma$ in the interior of a piece of paper.
He never published this (as ...
- 151k
6
votes
1
answer
604
views
When is the cut locus a finite tree?
Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the cut locus $C$ of $\Omega$ be the set of points $x \in \Omega$ for ...
- 5,038
31
votes
6
answers
2k
views
If a triangle can be displaced without distortion, must the surface have constant curvature?
Suppose $S$ is a Riemannian 2-manifold (e.g. a surface in $\mathbb{R}^3$).
Let $T$ be a geodesic triangle on $S$: a triangle whose edges are geodesics.
If $T$ can be moved around arbitrarily on $S$ ...
- 151k
6
votes
3
answers
365
views
Sliding through a curvature-bounded tube: Maximum volume?
My 1st question has a straightforward answer but I'd appreciate hints on a proof. My 2nd question is open from my point of view.
Q1. Is it the case that the maximum convex volume body inside a ...
- 151k
7
votes
1
answer
162
views
Estimate of number of boundary components of a compact Riemannian 2-surface
Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $-1$ and the diameter is at most $D$. Assume that near the boundary the ...
- 21.8k
7
votes
1
answer
231
views
Estimate of area of 2-dimensional surface
Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form ...
- 21.8k
5
votes
0
answers
464
views
Examples of spiraling geodesics?
Does there exist a closed, bounded surface $S$ embedded in $\mathbb{R}^3$
that has a geodesic $\gamma$ that spirals around a point $x$, getting closer
and closer, but never reaching $x$?
Here I ...
- 151k
5
votes
0
answers
391
views
Gage-Grayson-Hamilton curve-shortening flow, at an angle
The Gage-Grayson-Hamilton curve-shortening flows along the normal to the curve:
&...
- 151k
3
votes
0
answers
127
views
Behaviour of geodesics on surfaces as one of the two endpoints moves slightly
Let $u$ and $v$ be two points on a surface (I guess, a Riemann surface) $\Sigma$ such that there is a unique geodesic between $u$ and $v$ on $\Sigma$. Now let $l$ be an arbitrary line that passes ...
- 415
6
votes
1
answer
184
views
Self-avoiding/reflecting geodesics on a convex surface
Let $S$ be the surface of a convex body embedded in $\mathbb{R}^3$.
For me $S$ is a convex polyhedron,
but I am happy to view $S$ as a smooth body with positive Gaussian curvature
at each point, or ...
- 151k
29
votes
2
answers
1k
views
Is every closed curve in 3D a geodesic on a genus-0 surface?
Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$.
Q. Does there always exist a smooth, embedded, genus-zero surface
$S \subset \mathbb{R}^3$
such that $\gamma$ is a (...
- 151k
22
votes
4
answers
3k
views
What is the analog of the "Fundamental Theorem of Space Curves," for surfaces, and beyond?
The "Fundamental Theorem of Space Curves"
(Wikipedia link; MathWorld link)
states that there is a unique (up to congruence)
curve in space that simultaneously realizes
given continuous curvature $\...
- 151k
7
votes
4
answers
1k
views
On discrete version of curve shortening flow
One can define an analogous version of the curve shortening flow for polygons in $\mathbb R^2$, namely defined by the differential equation $\dot{p_i}(t)=\frac{v_i(t)}{|v_i(t)|^2}$, where $p_i$ is the ...
6
votes
5
answers
4k
views
Formulas for equidistant curves
I'm trying to draw on the computer a curve that keeps always the same distance(given as parameter) from a given curve. I know the formula for the given curve. I tried moving perpendicular to the first ...
- 351