RegularDiscretization of Triangle and Tetrahedron

[juliohm]

This PR adds methods for RegularDiscretization of Triangle and Tetrahedron. The parametrization of simplices of 2 or more dimensions is not orthogonal so we can't rely on RegularSampling.

I didn't add a test for Tetrahedron due to a missing isconvex method reported in #1205.

[codecov]

Codecov Report

Attention: Patch coverage is 85.71429% with 1 line in your changes missing coverage. Please review.

Project coverage is 88.43%. Comparing base (5245be5) to head (f77846e).
Report is 1 commits behind head on master.

Files with missing lines	Patch %	Lines
src/discretization/regular.jl	85.71%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1206      +/-   ##
==========================================
- Coverage   88.43%   88.43%   -0.01%     
==========================================
  Files         196      196              
  Lines        6159     6163       +4     
==========================================
+ Hits         5447     5450       +3     
- Misses        712      713       +1     

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[Copilot]

Pull Request Overview

This PR introduces RegularDiscretization support for 2D and 3D simplices by adding new dispatch methods and corresponding tests.

• Adds discretize overloads for Triangle and Tetrahedron using a generic discretizewithinbox helper.
• Updates existing polygon test resolution and adds a new triangle discretization test (tetrahedron tests pending isconvex support).
• Refactors fallback in src/discretization/regular.jl to call discretizewithinbox from both the general and specific dispatch paths.

Reviewed Changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated no comments.

File	Description
test/discretization.jl	Reduced polygon resolution, added triangle discretization test, pending tetrahedron test.
src/discretization/regular.jl	Introduced discretizewithinbox, added overloads for Triangle and Tetrahedron.

Comments suppressed due to low confidence (5)

test/discretization.jl:464

• Consider adding a placeholder or @test_throws for Tetrahedron once isconvex is implemented, to ensure coverage for 3D simplex discretization.

tri = Triangle(cart(0, 0), cart(1, 0), cart(0, 1))

src/discretization/regular.jl:30

• Add a docstring above this method to explain that triangle discretization is implemented by sampling its bounding box and then restricting to the triangle.

discretize(triangle::Triangle, method::RegularDiscretization) = discretizewithinbox(triangle, method)

src/discretization/regular.jl:32

• Add a docstring for the tetrahedron overload to clarify the approach and any limitations (e.g., requirement for a convex geometry).

discretize(tetrahedron::Tetrahedron, method::RegularDiscretization) = discretizewithinbox(tetrahedron, method)

src/discretization/regular.jl:34

• [nitpick] The helper name discretizewithinbox could be more idiomatically written as discretize_within_box to match snake_case conventions and improve readability.

function discretizewithinbox(geometry::Geometry, method::RegularDiscretization)

src/discretization/regular.jl:30

• Since the general discretize(Geometry, RegularDiscretization) fallback already calls discretizewithinbox, these explicit overloads for Triangle and Tetrahedron may be redundant; consider relying solely on the fallback unless specialized behavior is needed.

discretize(triangle::Triangle, method::RegularDiscretization) = discretizewithinbox(triangle, method)

[JoshuaLampert]

LGTM. I only left one minor suggestion below.

[JoshuaLampert]

Thanks!
Just FYI (probably you are already aware of it): Something similar probably has to be done for RegularSampling. I just tried on master

julia> using Meshes

julia> tri = Triangle(Point(0, 0), Point(1, 0), Point(0, 1))
Triangle
├─ Point(x: 0.0 m, y: 0.0 m)
├─ Point(x: 1.0 m, y: 0.0 m)
└─ Point(x: 0.0 m, y: 1.0 m)

julia> p = sample(tri, RegularSampling(10))
Base.Iterators.Flatten{Base.Generator{Tuple{Base.Generator{Base.Iterators.ProductIterator{Tuple{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}, Meshes.var"#459#463"{Triangle{𝔼{2}, CoordRefSystems.Cartesian2D{CoordRefSystems.NoDatum, Unitful.Quantity{Float64, 𝐋, Unitful.FreeUnits{(m,), 𝐋, nothing}}}}}}, Base.Generator{Tuple{}, typeof(identity)}}, typeof(identity)}}(Base.Generator{Tuple{Base.Generator{Base.Iterators.ProductIterator{Tuple{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}, Meshes.var"#459#463"{Triangle{𝔼{2}, CoordRefSystems.Cartesian2D{CoordRefSystems.NoDatum, Unitful.Quantity{Float64, 𝐋, Unitful.FreeUnits{(m,), 𝐋, nothing}}}}}}, Base.Generator{Tuple{}, typeof(identity)}}, typeof(identity)}(identity, (Base.Generator{Base.Iterators.ProductIterator{Tuple{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}, Meshes.var"#459#463"{Triangle{𝔼{2}, CoordRefSystems.Cartesian2D{CoordRefSystems.NoDatum, Unitful.Quantity{Float64, 𝐋, Unitful.FreeUnits{(m,), 𝐋, nothing}}}}}}(Meshes.var"#459#463"{Triangle{𝔼{2}, CoordRefSystems.Cartesian2D{CoordRefSystems.NoDatum, Unitful.Quantity{Float64, 𝐋, Unitful.FreeUnits{(m,), 𝐋, nothing}}}}}(Triangle((x: 0.0 m, y: 0.0 m), (x: 1.0 m, y: 0.0 m), (x: 0.0 m, y: 1.0 m))), Base.Iterators.ProductIterator{Tuple{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}((0.0:0.1111111111111111:1.0, 0.0:0.1111111111111111:1.0))), Base.Generator{Tuple{}, typeof(identity)}(identity, ()))))

julia> collect(p)
ERROR: DomainError with (1.0, 0.1111111111111111):
invalid barycentric coordinates for triangle.

So, this looks like a similar issue, which would be good to resolve.

[juliohm]

I think the RegularSampling method is not well-defined with simplices that have more than 1 parametric dimension due to the non-orthogonality of the barycentric coordinates. If we sample the first coordinate in a triangle, the second is defined by the sum-to-one constraint. The points wouldn't be distributed regularly inside the triangle, at least with our current definition of RegularSampling.

[JoshuaLampert]

Ah, I would have expected that RegularSampling would simply return (an iterator of) the points of RegularSampling in the bounding box that lie in the triangle.

[juliohm]

That is a good point. Maybe we should consider this more general definition!

[JoshuaLampert]

Sounds good.

[juliohm]

If we allow RegularDiscretization then it is indeed strange to not allow RegularSampling. I will open an issue to track this.

[JoshuaLampert]

Alternatively, one could think of providing a translation from $[0,1]^2$ to the barycentric coordinates like we do in MeshIntegrals.jl: https://github.com/JuliaGeometry/MeshIntegrals.jl/blob/main/src/specializations/Triangle.jl#L30. But I don't know how much sense this makes for sampling to be honest. But it would also make things easier if every parametrized geometry would be parametrized by $[0,1]^n$ and not just almost every. See also #1093 (comment).

[juliohm]

Yes, that inconsistency bothered me too in the past, but most algorithms I know that use triangles and tetrahedra end up requiring barycentric coordinates. The mapping of the box coordinates to the barycentric coordinates (and vice-versa) could be useful in other contexts besides numerical integration. The other examples with unbounded parametric coordinates like Line and Plane would also require some thought.