Simple Julia library, that defines some origin centered geometric objects (Ellipsoid, TruncatedSquarePyramid, Torus, among others).

The main functions here are ftransform(f, s::Shape), ptransform(s::Shape) and domain(s::Shape). The first one maps a function f(x, y, z) to another g(λ, μ, ν) * J(λ, μ, ν), where g(λ, μ, ν) is basically f(x(λ, μ, ν), y(λ, μ, ν), z(λ, μ, ν)) within the volume of a shape s but under a change of variables to a rectangular domain defined by (λ, μ, ν) and J(λ, μ, ν) is the Jacobian determinant of the transformation. The limits of the domain are given by domain(s).

ptransform(s::Shape) returns a function that maps a point (λ, μ, ν) on the domain given by domain(s) to a tuple (j, x, y, z), where p = (x, y, z) corresponds to the cartesian coordinates of a point inside s, and j is the is the Jacobian determinant of the transformation (x, y, z) ↦ (λ, μ, ν) evaluated on p.

ftransform(f, s) can be used together with an integration library, e.g. NIntegration.jl, to find out the integral of the function f within the region bounded by the surface of the shape s.

It also extends the Base method in to determine if a point in three dimensions lies within the volume defined by each object.

GeometricTransforms.jl should work on Julia 0.5 and later versions, and can be installed from a Julia session by running

julia> Pkg.clone("https://github.com/pabloferz/GeometricTransforms.jl.git")

Once installed, run

using GeometricTransforms

Let's see how the library can be used along an integration library to approximate the volume of a sphere of radius 1.

julia> using GeometricTransforms
julia> using BenchmarkTools

julia> Pkg.clone("https://github.com/pabloferz/NIntegration.jl.git")
julia> using NIntegration

julia> let
           r = 1.0
           S = Sphere(r)
           f = (x, y, z) -> 1.0
           integrand = (x, y, z) -> Point(x, y, z) in S
           xmin, xmax = (-r, -r, -r), (r, r, r)
           @btime nintegrate($integrand, $xmin, $xmax)
       end
  56.498 ms (752069 allocations: 16.04 MiB)
(4.189016214102217,0.1294570371126839,1000125,3938 subregions)

julia> let
           r = 1.0
           S = Sphere(r)
           f = (x, y, z) -> 1.0
           integrand = ftransform(f, S)
           xmin, xmax = domain(S)
           @btime nintegrate($integrand, $xmin, $xmax)
       end
  4.541 μs (46 allocations: 1.05 KiB)
(4.188790204114109,6.332978594815053e-7,127,1 subregion)

julia> let
           r = 1.0
           S = Sphere(r)
           f = (x, y, z) -> 1.0
           integrand = ftransform(f, S)
           xmin, xmax = domain(S)
           @btime nintegrate($integrand, $xmin, $xmax, reltol=1e-10)
       end
  62.387 μs (345 allocations: 9.31 KiB)
(4.188790204786391,7.700952182138001e-12,889,4 subregions)

julia> 4π / 3
4.1887902047863905

This work was financially supported by CONACYT through grant 354884.