Add tests for Ellipsoid and Hexahedron

[JoshuaLampert]

Integrating Ellipsoid and Hexahedron should simply work. However, I needed to set pretty high tolerances for Ellipsoid for some reason. Do you know why @mikeingold?
I also added other geometries, which do not implement any parametric function yet, in the support matrix, cf. #57

[JoshuaLampert]

Just as a note: The link behind the 🎗️ in the docs doesn't seem to work.

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[JoshuaLampert]

However, I needed to set pretty high tolerances for Ellipsoid for some reason. Do you know why @mikeingold?

Ah, I see why. The result of measure is already off by the tolerance because Meshes.jl doesn't implement an analytical measure for an Ellipsoid (which is also not so easy to determine in general) and thus uses a fallback.

julia> using Meshes, MeshIntegrals

julia> origin = Point(0, 0, 0)
Point with Cartesian{NoDatum} coordinates
├─ x: 0.0 m
├─ y: 0.0 m
└─ z: 0.0 m

julia> radii = (1.0, 1.0, 1.0)
(1.0, 1.0, 1.0)

julia> ellipsoid = Ellipsoid(radii, origin)
Ellipsoid
├─ radii: (1.0 m, 1.0 m, 1.0 m)
├─ center: Point(x: 0.0 m, y: 0.0 m, z: 0.0 m)
└─ rotation: LinearAlgebra.UniformScaling{Bool}(true)

julia> using Unitful: m

julia> 4pi*m^2 - measure(ellipsoid)
0.02225157793214194 m^2

julia> 4pi*m^2 - integral(f, ellipsoid, GaussLegendre(100))
-4.96278573791642e-11 m^2

The result from integral is actually much more precise.

[mikeingold]

Ah, I see why. The result of measure is already off by the tolerance because Meshes.jl doesn't implement an analytical measure for an Ellipsoid (which is also not so easy to determine in general) and thus uses a fallback.

The result from integral is actually much more precise.

Exactly. This is something I've definitely seen. When there's no analytic solution defined for measure, Meshes will simplexify the geometry and essentially take a finite element approximation, so integrating a unit integrand function will probably be more accurate. This is one of the motivations behind moving away from the old auto tests that relied on measure as a benchmark.

[JoshuaLampert]

Once the overall goal to test against analytical solutions for non-trivial functions is achieved in the future, we will not depend on measure anyway.

[mikeingold]

Just as a note: The link behind the 🎗️ in the docs doesn't seem to work.

Weird. It looks like the Documenter site's `stable` version is stale and hasn't been updated since ~v13.2. The pages under `dev` are up to date, though. Have you ever seen this kind of thing happen before?

…

On Wed, Oct 9, 2024 at 1:04 PM Joshua Lampert ***@***.***> wrote: Once the overall goal to test against analytical solutions is achieved in the future, we will not depend on measure anyway. — Reply to this email directly, view it on GitHub <#102 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AA657OWQVEMHK4B7RCK6HKDZ2VOYHAVCNFSM6AAAAABPUZPOG6VHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHMZDIMBSHA2TGOJVGQ> . You are receiving this because you were mentioned.Message ID: ***@***.***>

[JoshuaLampert]

Weird. It looks like the Documenter site's stable version is stale and
hasn't been updated since ~v13.2. The pages under dev are up to date,
though. Have you ever seen this kind of thing happen before?

Indeed. I don't know why this is and I don't see anything suspicious in the logs.

[mikeingold]

Weird. It looks like the Documenter site's stable version is stale and
hasn't been updated since ~v13.2. The pages under dev are up to date,
though. Have you ever seen this kind of thing happen before?

Indeed. I don't know why this is and I don't see anything suspicious in the logs.

We don't have to hold up this PR over it. I just added #103 for tracking.