Point Cloud Differential Geometry

struct Manifold{T <: Real}
    mesh :: Mesh{T}
    surf :: Surface{T}
end

Store the representation of a differential geometry object. mesh is the empirical point cloud. surf is the estimated differentiable surface.

struct Mesh{T <: Real}
    r   :: Array{T, 2}
    vₙ  :: Array{T, 2}
    tri :: Array{Int, 2}
end

Store a 2 dimensional triangular mesh, embedded into arbitrary dimensions. r and vₙ denote vertex positions and normals respectively. tri denotes (non-oriented) triangular faces.

struct Surface{T <: Real}
    Θ  :: Array{T, 2}
    x  :: Array{T}
    L  :: T
    Λ  :: Array{Polynomial{T}}
    ∂Λ :: Array{Polynomial{T}}
end

Store a representation of a 2D surface embedded into higher dimensional Euclidean space. Fits the surface by:

1. Partition the x axis so that each resultant interval has, on average, 20 points.
2. Fit points within each partition to a 2D ellipse.
3. Fit a polynomial function to each elliptical parameter over all partitions.
4. Use polynomials to estimate tangent vectors.

Θ denotes the elliptical parameters fit per partition. x denotes the input data. L denotes the length along the x axis. Λ denotes the polynomials for each elliptical parameter. ∂Λ denotes the derivative of polynomials for each elliptical parameter.

function basis(s::Surface, r)

Compute the tangent vectors $\hat{\bm{e}}_\phi, \hat{\bm{e}}_x$ associated to each point within r. Assumes all points within r are distributed over the surface. Assumes r is sized $N \times 3$.

ellipse(r)

Fit an ellipse to 2D point cloud r. r is assumed to be sized $N \times 2$.

function interpolate(M, ϕ, r)

Interpolate the tensor field ϕ, defined at points r onto the vertices of manifold M. Interpolation is computed by finding the containing triangle within the mesh of M for each point r. Linear interpolation is performed per face.

makefield(ℳ::Manifold, ϕ::AbstractArray; field::Symbol = :ℝ³)

Return an arbitrary tensor field defined over either the embedding space or the cylindrical pullback of manifold M. Higher level function than either scalar or vector.

mesh(io::IO, type::Symbol=:obj)

Load a Mesh object from stream io formatted with type. As of now, only .obj files are supported.

function order(tri, r)

Reorder the triangulation so that all labels are counterclockwise.

function pullback(ℳ::Manifold)

Returns the 2D cylindrical projection of the mesh, as estimated by the internal surface.

function pullback(s::Surface, r)

Transform Euclidean point cloud r into a 2D cylindrical projection defined by s. Cylindrical coordinates are [x,φ]. Assumes all points within r are distributed over the surface. Assumes r is sized $N \times 3$.

rescale(x)

Rescale array x to run between $[0,1]$.

scalar(M::Manifold, ϕ, field::Symbol)

Return a scalar field ϕ interpolated onto either the embedded space of manifold M or the pullback. If field is :ℝ² the pullback is computed. If field is :ℝ³ the embedding space is returned.

function triangulation(r₀, rᵢ)

Return the triangle containing each point given in rᵢ. The triangulation used is delaunay triangulation defined by point cloud r₀.

scalar(M::Manifold, ϕ, field::Symbol)

Return a vector field field ϕ interpolated onto either the embedded space of manifold M or the pullback. If field is :ℝ² the pullback is computed. Only the tangent space component is kept. If field is :ℝ³ the embedding space is returned.