Some geometry

Voronoi tasselation

The Voronoi tasselation consists in partitioning the plane into regions based on the distance to points in a specific subset of the plane.

For each point, there is a region of all points closer to it than any other one (Voronoi cells).

Simplex

A simplex is the generalisation of a triangle/tetrahedron to an arbitrary number of dimensions. The k simplex is a k-dimensional polytope, the convex hull of its k+1 vertices.

Given pointsu0,,ukRku_0, \ldots, u_k \in \mathbb{R}^k, withu1u0,,uku0u_1 - u_0, \ldots, u_k - u_0linearly independent, the simplex determined by them is the set of points

C:θ0u0++θkukθi0,0ik,i=0kθi=1C : {\theta_0 u_0 + \cdots + \theta_k u_k | \theta_i \geq 0, 0 \leq i \leq k, \sum_{i=0}^k \theta_i = 1}
  • the 2-simplex is a triangle;

  • the 3-simplex is a tetrahedron;

  • the 4-simplex is a 5-cell

References