All Questions
Tagged with computational-geometry dg.differential-geometry
18 questions
1
vote
0
answers
76
views
Translate of a geodesic that goes through a fixed point on $\mathbb{H}$
Consider the complex upper half plane $\mathbb{H}$ with the hyperbolic geometry. Fix a point $z \in \mathbb{H}$ and also a geodesic $c$. I want to find a hyperbolic translation $\gamma c$ passes that ...
- 577
0
votes
1
answer
175
views
Reference request: How to construct a diffeomorphism between point clouds
I'm interested in the following question:
Given two sets $S = \{x_1, ..., x_N\}$ and $T = \{y_1, ..., y_N\}$ each consisting of $N$ distinct points in $\mathbb{R}^n$, how can we construct a ...
- 637
3
votes
2
answers
157
views
Finding a point inside a surface
I have a triangulation of a surface without boundary in $\mathbb{R}^3$.
The triangulation gives a unit normal pointing outwards for each triangle. I need to find some point in the interior of the ...
- 33
1
vote
0
answers
87
views
On vertices on convex smooth closed surfaces
Question: Given a smooth, convex, closed surface. How would one find points on it with the property: the average of the geodesic distances from that point to all other points on the surface (the ...
- 5,979
2
votes
1
answer
223
views
Discrete curve-shortening flow – numerical implementation
I need to investigate the properties of open curves which evolve according to the standard curve-shortening flow (Wikipedia link), but with fixed extremes as boundaries (si it should converge to the ...
- 31
8
votes
1
answer
316
views
What is the form of the $(v_0,v_1)$-pizza curve?
Assume that there are two (competing) pizza houses situated at the points $0$ and $1$ on the complex plane. These pizza houses can deliver pizza to points of the plane with the largest velocities $v_0$...
- 41.9k
2
votes
2
answers
399
views
Computer algebra for calculating curvature when the tensor metric is very big
Is there a computer algebra method to compute the curvature of a Riemannian metric on the plane when the metric tensor has long entries $E,F,G$
The computation by hand is very ...
- 356
5
votes
0
answers
1k
views
Reach of manifold vs. $C^k$-manifold
The reach $\tau_M$ of a manifold $M$ is the largest number such that any point at distance less than $\tau_M$ from $M$ has a unique nearest point on $M$.
This concept seems quite related to the local ...
- 151k
0
votes
1
answer
124
views
Triangle inside the Closed Curve
For any piece wise smooth, simple closed curve $\gamma$ in the Euclidean plane $E^2$ and fix a point $G$ inside the area circled by $\gamma$.
Show: There exists three points $A,B$ and $C$ on the $\...
- 1,915
6
votes
1
answer
148
views
Does a minimum area disk that is bounded by a cycle $C$ continuously deform in $R^3$ as $C$ moves in $R^3$?
Let $C_1=(v_1,v_2,\ldots,v_{i-1},v_i)$ and $C_2=(v_1,v_2,\ldots,v_{i-1},v'_i)$ be two cycles that are drawn in $R^3$ in the shape of an unknot (not knotted) with straight line segments as their edges (...
- 415
0
votes
0
answers
108
views
conformal deformation with fixed boundaries
For a flat plane with certain boundary, e.g., a rectangular patch, is it possible to conformally displace or deform such patch to a curved bump with exact same boundary? In this thesis, Dr. Keenan ...
- 109
1
vote
0
answers
91
views
Representing a Pullback as an Infinite Matrix
Let $M$ and $N$ be manifolds and let $T: M \to N$ be a bijective map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) be the space of all functions from $M$ (resp. $N$) to $\...
- 61
2
votes
1
answer
586
views
Inverse Problem for Pullback
Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) denote the space of smooth functions from $M$ (resp. $N$) ...
- 61
8
votes
1
answer
1k
views
Dubins car shortest paths: Decidable?
A Dubins car follows a
Dubins path
in $\mathbb{R}^2$, with constant wheel speed and
limited turning radius.
It is known that the shortest Dubins path in the absence
of obstacles follows circular
arcs ...
- 151k
11
votes
5
answers
1k
views
Segments of Voronoi Diagrams on smooth manifolds. Are they geodesics?
Let $S$ be a patch of a smooth 2-manifold in $\mathbb{R}^3$, and pick two distinct points $a,\ b \in S$. Let $c$ be the set of points on $S$ equidistant to $a$ and $b$, where distance is defined by ...
- 273
8
votes
2
answers
362
views
Computational methods for dealing with geometrically complicated solid boundaries in fluid-air interface problems
Hello,
I am a PhD student who does not have extensive computational experience seeking advice from those experienced with computational modelling as to which method would be most appropriate for ...
- 147
4
votes
3
answers
919
views
Area-preserving map between rectangles and fat polygons
Are there any well-known continuous area-preserving maps between fat convex polygons and fat rectangles? Specifically, given a fat convex polygon $C$, is there a natural way to choose a fat rectangle ...
- 1,125
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