Bernstein basis functions for simplices

p = barycentric2cartesian(s, λ)

Convert barycentric to Cartesian coordinates.

Arguments

• s: Simplex vertices in Cartesian coordinates. s has N ≤ D + 1 vertices in D dimensions.
• λ: Point in barycentric coordinates

Result

• p: Point in Cartesian coordinates

bernstein(α, λ)
bernstein(s, α, x)

Evaluate Bernstein polynomial

The order of approximation is given implicity by sum(α).

Arguments

• s: Vertices of simplex
• α: multi-index describing polynomial
• x: location where Bernstein polynomial is evaluated in Cartesian coordinates
• λ: location where Bernstein polynomial is evaluated in barycentric coordinates

Overlap integrals between Bernstein polynomials (which are neither orthogonal nor normalized)

λ = cartesian2barycentric(s, p)

Convert Cartesian to barycentric coordinates.

Arguments

• s: Simplex vertices in Cartesian coordinates. s has N ≤ D + 1 vertices in D dimensions.
• p: Point in Cartesian coordinates

Result

• λ: Point in barycentric coordinates

cartesian2barycentric_setup(s)

Prepare to convert Cartesian to barycentric coordinates.

The returned setup structure can be passed to cartesian2barycentric instead of the simplex vertices. This pre-calculates certain operations and is more efficient.

Arguments

• s: Simplex vertices in Cartesian coordinates. s has N=D+1 vertices in D dimensions.

Result

• setup