All Questions
Tagged with curves-and-surfaces mg.metric-geometry
37 questions
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
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
26
votes
1
answer
846
views
Disc bounded by a plane curve
Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$.
Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve?
It is easy to find an open disc ...
- 45k
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
19
votes
0
answers
841
views
I found a (probably new) family of real analytic closed Bezier-like curves; is it publishable?
Given $n$ distinct points $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)$ in the plane $\mathbb{R}^2$, I associate a real analytic map:
$f_{\mathbf{x}}: S^1 \to \mathbb{R}^2$
with the following ...
- 5,215
15
votes
0
answers
517
views
Functions approximated by rolling epicycle curves
Imagine a decreasing sequence of (positive) radii $r_1 > r_2 > r_3 > \cdots$
and a series of nested circles $C_1 \supset C_2 \supset C_3 \supset \cdots$
with these radii,
initially each ...
- 151k
13
votes
0
answers
254
views
Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$
An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane.
The following questions are motivated by Anton Petrunin's Disc bounded ...
- 7,276
8
votes
2
answers
287
views
Does the surface area of the unit Lp ball go to zero for all $p < \infty$?
We know about volume: The $L_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of ...
- 4,529
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 ...
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
7
votes
0
answers
102
views
Is there a convex three-dimensional body with constant width and only finitely-many equilibria? Or: do spheroform gömböcök exist?
Mathematical questions. The mathematical (and 'gravity'-free) formulation of the question in the title is given by the following questions:
Q1. Does there exist $(a,b)\in\omega^2\setminus\{(0,0)\}$ ...
- 6,051
7
votes
0
answers
410
views
Can generalization of a generalization Pascal theorem, Pappus theorem to Higher Dimensions? [closed]
Please see a chain of six circles associated with a conic. This is a generalization of Pascal theorem, Pappus theorem. I reformulate as following:
Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a ...
- 477
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
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
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
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
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
5
votes
1
answer
395
views
Average distance to a curve of fixed length
Let $C$ be a continuous curve in the unit square having length $L$. Is there a lower bound on the average distance between the points in the unit square and $C$, as a function of $L$? Is there an ...
- 1,125
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
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
3
votes
2
answers
3k
views
Parametrization of the intersection of an ellipsoid with a sphere
First I would like to say that geometry is far away from my domain.
I have encountered a problem that has a geometric formulation and I don't even know if this is a difficult or an easy problem.
So ...
- 1,334
3
votes
0
answers
146
views
Chord of fixed length traveling around a Jordan curve
Let $C$ be a Jordan curve with nice enough properties whenever necessary (e.g. smooth, or just rectifiable, perhaps). I am interested in knowing how long can a chord be that "traverses" the ...
- 1,763
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
3
votes
0
answers
310
views
Does a rectangle exist on any Jordan curve?
Let $C$ be a Jordan curve in $\mathbb{R}^2$. Does there exist points $P,Q,R,S$ on $C$ such that quadrangle $PQRS$ is a non-degenerate rectangle?
user80247
2
votes
1
answer
139
views
The radius of an interval's image through a space-filling curve
Take $f:[0,1]\to [0,1]^n$ a continuous tour around $[0,1]^n,$ say, some iteration of a Hilbert curve. For $\varepsilon \in (0,1)$ what is the following thing called and are there any nontrivial upper ...
2
votes
0
answers
52
views
Efficiently determining surface intersections along a line segment
Background
In general, I know how to determine the points of intersection between a surface and a line. In my case, I may have a large number of defined surfaces that may (or may not) intersect each ...
- 121
2
votes
0
answers
166
views
Pascal theorem for three dimensions
A year ago I found the Pascal theorem for three dimentions as follows:
Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$,...
- 1,776
1
vote
1
answer
651
views
When are principal lines of curvature geodesics?
Let $S$ be a smooth surface embedded in $\mathbb{R}^3$.
When are (some of) the principal lines of curvature geodesics
on $S$? Perhaps on the ellipsoid below, the (blue) central
cycle, a max principal ...
- 151k
1
vote
0
answers
97
views
A third degree surface and a touching sphere [closed]
Let consider a surface $z=1/(xy)$ and a sphere defined by $(x-1.5)^2+(y-1.5)^2+(z-1.5)^2=3/4$. The sphere touches the surface at (1,1,1). Is it possible to prove that point (1,1,1) is the only ...
- 1,550
1
vote
0
answers
109
views
Intersection points of closed curves inscribed in a convex polygon
Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most ...
- 31
0
votes
2
answers
129
views
Planar curves identical to their inverses
Is the right strophoid
the only planar curve $C$ whose inverse curve w.r.t. some circle (in this case: centered on the origin)
is identical to $C$?
&...
- 151k
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
0
votes
0
answers
95
views
Pushing figures into holes
Let $\gamma_1,\gamma_2:[0,1]\to \mathbb{R}^2$ - smooth curve, $\gamma_i(0)=\gamma_i(1)$, $X_1$ and $X_2$ are the areas bounded by the corresponding curves. . Suppose we have an $X_1 $-shaped hole, and ...
- 107
-1
votes
1
answer
467
views
Meeting point of the vertices of a square cloth on x-y plane [closed]
Consider a standard square sheet lying on the xy plane with edge length n. Is it possible to determine the coordinates (x, y, z) of the point where the vertices of the sheet will meet, when each of ...
- 101
-2
votes
1
answer
587
views
Is the conjecture true for n-sphere $(n>2)$? [closed]
This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here:
PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...
- 1,776