Verified the existence with method I described in Grassmann.jl A\b 3x faster than Julia's StaticArrays.jl

First, you need to convert the mesh into `Grassmann`

format:

```
using Grassmann
function mesh_init(m::GeometryBasics.Mesh)
c,f = coordinates(m),faces(m)
s = size(eltype(f))[1]; V = SubManifold(ℝ^s)
p = ChainBundle([Chain{V,1}(1.0,k...) for k ∈ c])
t = ChainBundle([Chain{p,1}(SVector(k)) for k ∈ f])
return (p,t)
end
```

Then define the following `checkpoint`

for determining the tetrahedron containing a point:

```
function checkpoint(P::Chain{V,1},t) where V
p = Manifold(t)
for i ∈ 1:length(t)
P ∈ Chain{V,1}(p[value(t[i])]) && (return i)
end
return 0
end
```

Then you can determine whether there exists a tetrahedron with point `P`

or not (returns 0 if not).

```
julia> p,t = mesh_init(result) # result from TetGen
(Λ¹⟨++++⟩×211, Λ¹Λ¹⟨++++⟩×211×566)
julia> checkpoint(Chain{SubManifold(ℝ^4),1}(1,2,2,2),t)
122
```

Note that I am using homogeneous coordinates with (1,x,y,z) coordinates for the points, so you will have to use 4 dimensional points with an additional 1 coordinate.

You will find that indeed the mesh data contains simplices having points in the problematic regions.

PS, it would be great if `Makie`

could natively support `Grassmann`

so I don’t need `GeometryBasics`

as a compatibility layer. Instead of `GeometryBasics.Point`

I prefer using `Grassmann.Chain{V,1}`

, because I need that type in order to do my calculations. As you can see, I am using homogeneous coordinates with `Chain{V,1}`

types in `Grassmann`

to represent mesh data. Currently, I need to convert this mesh data back into `GeometryBasics.Point`

for plotting (using cache), which works fine, but I would much prefer being able to work with `Grassmann.Chain{V,1}`

directly without conversions.

```
GLMakie.mesh(t, color=(:blue, 0.1), transparency=true)
wireframe!(ans[end][1])
```

As you can see, it’s possible to use `mesh`

function on `Grassmann`

data as well, but it requires the conversion process with caching I mentioned above. I’d prefer to have `Makie`

work with `Grassmann`

point data directly instead of converting to `GeometryTypes.Point`

.