# Tagged Questions

**4**

votes

**2**answers

202 views

### What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space?

The question is mostly in the title.
Imagine I have some non-convex polyhedron $P$, and I would like to find the smallest diameter ring that it can pass through in 3-space, undergoing any necessary ...

**2**

votes

**2**answers

461 views

### Threading pinholes in the wall of cylinder to pass through an internal coordinate

Imagine I take a sheet of paper and use a pin to generate an $N$x$M$ rectangular array of small holes. I then fold the sheet to form a cylinder of radius $r_c$ and height $h_c$, where there are $N$ ...

**1**

vote

**1**answer

399 views

### Decomposing a sphere (or defomed sphere) into a vertex-transitive graph with fixed-length curved edges

Please see the original problem specification (which Joseph O'Rourke was responding to in his answer) below.
Motivation -
I'm interested in a particular case of the problem where one wants to ...

**1**

vote

**1**answer

258 views

### Constructing a graph that approximates a sphere using rotationally symmetric building blocks with equal numbers of edges

I'd like to construct a graph that approximates a sphere in 3-space, but I'm placed under the following constraints:
(1) - I am only allowed to use a construction block, $v_i$, consisting of a single ...

**1**

vote

**1**answer

128 views

### Inferring geometric properties of a polytope from intersection volumes of spheres at unknown coordinates on its surface

Let's say we have some polytope $P$ in 3-space (which is not necessarily convex) as well as some number of points on its surface, $(g_1, ..., g_N)$. We are provided no information about the ...

**3**

votes

**5**answers

572 views

### Is the following two-dimensional graph likely to be globally rigid?

Consider the two-dimensional non-planar graph $G$, with known topology and edge lengths $(r_1, r_2, ... r_N) \in R$, but unknown vertex coordinates. We further specify that:
All vertices within a ...

**4**

votes

**2**answers

311 views

### Approximate search space on a 5x5x5 cube with 3 different possible classes?

Hey all,
I read the meta, and I realize this question might be pretty elementary for this site, but I'm having trouble computing this, and I know it won't take too much insight for someone to give me ...

**1**

vote

**0**answers

370 views

### Catmull-Clark Subdivision and weights

I've been toying with a Catmull-Clark subdivision algorithm; I also compared my results to results I found online (e.g. images.google.com).
The weights used to relocate the old vertices of the mesh ...

**2**

votes

**3**answers

317 views

### Can we uniquely define a graph to have the topology of a polytope via proper edge length selection?

I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure ...

**4**

votes

**3**answers

955 views

### Optimal packing of spheres tangent to a central sphere

Please consider a central, ordinary 2-sphere $S_1$, of some radius $r_1$, and a second ordinary sphere, $S_2$, of radius $r_2$, where $r_2 \leq r_1$.
My question concerns optimal values for the ...

**8**

votes

**1**answer

843 views

### Expected number of steps for a discrete random walk to visit every point on an N-dimensional rectangular lattice

Please imagine a discrete random walk on an N-dimensional rectangular lattice with dimensional lengths $(l_1, ..., l_N) \in L$ and total lattice points $P = \prod{l_i}$, for $i = 1, ..., N$. At each ...

**1**

vote

**4**answers

1k views

### Finding the union of N random circles arbitrarily (or conspiratorially) placed on a two-dimensional surface

Please consider a two-dimensional surface populated with a set of Cartesian coordinates $(x_i, y_i)$ for $N$ circles with individual radii $r_i$, where $r_{min} < r_i < r_{max}$.
Here, the ...

**1**

vote

**2**answers

767 views

### Calculating the surface area distribution of two-dimensional projections for a polytope

My question concerns the existence of a nice (deterministic?) method/algorithm for calculating the distribution of surface areas for two-dimensional projections of an arbitrary polytope (or convex ...

**9**

votes

**4**answers

655 views

### Characterizing a tumbling convex polytope from the surface areas of its two-dimensional projections

My general question concerns what we can learn about an arbitrary, three-dimensional convex polytope (or convex hull of an arbitrary polytope) strictly from the surface areas of its two-dimensional ...

**9**

votes

**2**answers

451 views

### When can a freely moving sphere escape from a 'cage' defined by a set of impassible coordinates?

To ask this question in a (hopefully) more direct way:
Please imagine that I take a freely moving ball in 3-space and create a 'cage' around it by defining a set of impassible coordinates, $S_c$ ...

**7**

votes

**2**answers

555 views

### Deciding membership in a convex hull

Problem: Given points $u,v_1,\dots,v_n\in\mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1,\dots,v_n.$
This can be done efficiently by linear programming (time polynomial in ...