**5**

votes

**2**answers

1k views

### Light rays bouncing around inside a sphere in d-dimensions

Suppose $S=\mathbb{S}^d$ is a unit sphere in $(d-1)$ dimensional space, with $d=3$ of special interest.
The surface of $S$ is a perfect (internal) mirror.
You stand at point $x$ (not the sphere center ...

**5**

votes

**3**answers

320 views

### What is the most general class of metric spaces for which the closest pair of points in a finite subset can be found in time O(n^(1+eps))?

What is the most general class of metric spaces for which the closest pair of points in any finite subset can be found in time O(n^(1+eps))? I have studied how to do this in O(n log(n)) in the plane, ...

**5**

votes

**1**answer

636 views

### Minimizing variance of distances between points when mean distance is fixed

In Rd, I have n > d+1 points. The mean distance between pairs of points is 1. How can I minimize the variance of the distances (equivalently, the mean squared distance)? I'm mainly interested in d ...

**6**

votes

**1**answer

885 views

### Algorithm for embedding a graph with metric constraints

Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide if there is an ...

**2**

votes

**2**answers

1k views

### Computational geometry, tetrahedron signed volume

Short version: I'm trying to compute the orientation of a triangle on a plane, formed by the intersection of 3 edges, without explicitly computing the intersection points.
Long version: I need to ...

**1**

vote

**0**answers

177 views

### Finding equations for projective bundles associated to vector bundles over explicitly given varieties

Suppose I have a projective variety V over a field k. It is given explicitly in terms of homogeneous equations. Moreover, say I have an explicitly given vector bundle E (in terms of a module ...

**4**

votes

**1**answer

365 views

### Computing places over x in F/K(x)

Let $F$ be a function field of "transcendental degree one" over its full constant field $K$. Let $x \in F \backslash K$. We know the divisor of $(x) = (x) - (1/x)$ in $K(x)$. Could you please give me ...

**4**

votes

**1**answer

634 views

### Finding integer points on an N-d convex hull

Suppose we have a convex hull computed as the solution to a linear programming problem (via whatever method you want). Given this convex hull (and the inequalities that formed the convex hull) is ...

**3**

votes

**1**answer

4k views

### split polygon into minimum amount of rectangles and triangles

Hi
is there an algorithm which cuts a polygon into a minimum amount of preferably rectangles and where not possible (e.g. edges) into triangles?

**2**

votes

**3**answers

416 views

### find the collision of a particle with a swept triangle.

Given there is triangle: V in 3D space that transforms over time t -> t1 to V1, and a static point P is somewhere in 3d space, how can I determine if P ever collides with V, and if so at what value of ...

**6**

votes

**4**answers

883 views

### How to compute the average distance till intersection within a triangle in R^2?

Lots of simple questions because I am a noob.
You are given 3 points in R^2; A, B, C forming a triangle with area > 0. You pick an arbitrary point inside ABC and an arbitrary direction. After some ...