CanonicalSimplex {SimplicialCubature}

Internal functions for defining/working with simplices.

Description

These are utility functions that are useful when defining/working with simplices in n-dimensional space.

Usage

CanonicalSimplex(n)
UnitSimplexV(n)
SimplexVolume(S)
SimplexSurfaceArea(S3)
JacobianS2Canonical(S2)

Arguments

n

positive integer giving the dimension of the space

S

an n x (n+1) matrix specifying a single n-dimensional simplex; the columns S[,1],..,S[,n+1] give the vertices of the simplex.

S2

an n x (m+1) matrix specifying a single m-dimensional simplex, with m <= n; the columns S2[,1],..,S2[,m+1] give the vertices of the simplex.

S3

an n x n matrix specifying a single (n-1)-dimensional simplex inside n-dimensional space; the columns S3[,1],..,S3[,n] give the vertices of the simplex.

Value

Let e[j] be the j-th standard unit basis vector. CanonicalSimplex(n) gives the simplex with columns being vertices of the canonical simplex in n-dimensions: the n-dim. simplex with vertices (0,0,...,0) and e[1],...,e[n]. A vector (u[1],...,u[n]) is in the canonical simplex if 0 <= u[i] <= 1 for all i and sum(u) <= 1. UnitSimplexV(n) gives the vertices (V-representation) of the unit simplex, namely e[1],...,e[n]. A vector (u[1],...,u[n]) is in the unit simplex if 0 <= u[i] <= 1 for all i and sum(u) == 1. SimplexVolume(S) returns the n-dim. volume of S and SimplexSurfaceArea(S3) computes the (n-1)-dim. surface area of S3. JacobianS2Canonical(S2) returns the Jacobian of the transformation from an m-dim. simplex S2 to the m-dim. canonical simplex.

Examples

CanonicalSimplex(3)
UnitSimplexV(3)
SimplexVolume( CanonicalSimplex(3) ) 
SimplexSurfaceArea( UnitSimplexV(3) ) 
JacobianS2Canonical( UnitSimplexV(3) )