Bernstein.jl

Calculate Bernstein polynomials
Author eschnett
Popularity
2 Stars
Updated Last
8 Months Ago
Started In
May 2020

Bernstein Polynomial Basis

  • GitHub: Source code repository
  • GitHub CI

The Bernstein polynomials form a basis for polynomials living on simplices. This package calculates the Bernstein polynomials for simplices of arbitrary dimension.

This package also provides conversion functions between Cartesian and barycentric coordinates.

Examples

Convert between Cartesian and barycentric coordinates

julia> using StaticArrays

julia> using Bernstein

julia> # Define a triangle
       s = rand(SMatrix{3,2})
3×2 SArray{Tuple{3,2},Float64,2,6} with indices SOneTo(3)×SOneTo(2):
 0.814346  0.297149
 0.519781  0.620776
 0.345743  0.733385

julia> # Choose a point
       p = rand(SVector{2})
2-element SArray{Tuple{2},Float64,1,2} with indices SOneTo(2):
 0.1483582649665245
 0.3923504628863179

julia> # Convert to barycentric coordinates
       λ = cartesian2barycentric(s, p)
3-element SArray{Tuple{3},Float64,1,3} with indices SOneTo(3):
   3.523588891718682
 -10.621534570787583
   8.097945679068905

julia> # Convert back
       q = barycentric2cartesian(s, λ)
2-element SArray{Tuple{2},Float64,1,2} with indices SOneTo(2):
 0.14835826496652604
 0.3923504628863199

julia> p  q
true

You can also pass the simplex vertices as a vector of vectors SVector{N, SVector{D, T}} instead of a matrix.

If you convert many Cartesian to barycentric coordinates, then part of the transformation can be pre-calculated to increase efficiency. Call cartesian2barycentric_setup for this:

julia> using StaticArrays

julia> using Bernstein

julia> # Define a triangle
       s = rand(SMatrix{3,2})
3×2 SArray{Tuple{3,2},Float64,2,6} with indices SOneTo(3)×SOneTo(2):
 0.814346  0.297149
 0.519781  0.620776
 0.345743  0.733385

julia> # Choose a point
       p = rand(SVector{2})
2-element SArray{Tuple{2},Float64,1,2} with indices SOneTo(2):
 0.1483582649665245
 0.3923504628863179

julia> # Pre-calculate part of the transformation
       setup = cartesian2barycentric_setup(s);

julia> # Convert to barycentric coordinates
       λ = cartesian2barycentric(setup, p)
3-element SArray{Tuple{3},Float64,1,3} with indices SOneTo(3):
   3.523588891718682
 -10.621534570787583
   8.097945679068905

Evaluate Bernstein polynomials

You can evaluate Bernstein polynomials from barycentric coordinates or from Cartesian coordinates:

julia> using StaticArrays

julia> using Bernstein

julia> # Define a triangle
       s = rand(SMatrix{3,2})
3×2 SArray{Tuple{3,2},Float64,2,6} with indices SOneTo(3)×SOneTo(2):
 0.814346  0.297149
 0.519781  0.620776
 0.345743  0.733385

julia> # Choose a point
       p = rand(SVector{2})
2-element SArray{Tuple{2},Float64,1,2} with indices SOneTo(2):
 0.1483582649665245
 0.3923504628863179

julia> # Convert to barycentric coordinates
       λ = cartesian2barycentric(s, p)
3-element SArray{Tuple{3},Float64,1,3} with indices SOneTo(3):
   3.523588891718682
 -10.621534570787583
   8.097945679068905

julia> # Choose polynomial index and order
       # (The order is the sum of all coefficients)
       α = SVector(2,0,0);

julia> bernstein(α, λ)
3.523588891718682

julia> bernstein(s, α, p)
3.523588891718682

julia> bernstein(setup, α, p)
3.523588891718682

References

  • Douglas N. Arnold, Richard S. Falk, Ragnar Winther, "Geometric decompositions and local bases for spaces of finite element differential forms", 10.1016/j.cma.2008.12.017, arXiv:0806.1255 [math.NA].

  • Tom Lyche, Karl Scherer, "On the p-norm condition number of the multivariate triangular Bernstein basis", DOI: 10.1016/S0377-0427(00)00383-6.

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