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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 points

$u_0, \ldots, u_k \in \mathbb{R}^k$

, with$u_1 - u_0, \ldots, u_k - u_0$

linearly independent, the simplex determined by them is the set of points$C : {\theta_0 u_0 + \cdots + \theta_k u_k | \theta_i \geq 0, 0 \leq i \leq k, \sum_{i=0}^k \theta_i = 1}$

- the
*2-simplex*is a triangle; - the
*3-simplex*is a tetrahedron; - the
*4-simplex*is a 5-cell

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Voronoi tasselation

Simplex

References