DiscreteCellKForm{N,A}

A discrete k-form of grade N, represented by an indexible container A ( usually a Vector ) A 0-form (N==0) is a k-form with a value per dual face ( point at circumcentre, on cell boundary ) A 1-form (N==1) is a k-form with a value per dual edge ( boundary of one ring cell ) A 2-form (N==2) is a k-form with a value per dual vertex ( one ring cell around vertex )

DiscreteKForm{N,A}

A discrete k-form of grade N, represented by an indexible container A ( usually a Vector ) A 0-form (N==0) is a k-form with a value per vertex A 1-form (N==1) is a k-form with a value per edge A 2-form (N==2) is a k-form with a value per face

apply(𝑑̂ₐ, α, P, k)

Apply discrete differential operator 𝑑̂ₐ to k-form α at points P with k-vector k.

apply(α, u, k)

Apply differential k-form α at point u to k-vector k. Point u is a 1-vector.

apply(α, u)

Apply differential 0-form α at point u. Point u is a 1-vector.

Constant 0-form. Same value at any point

apply(𝑑̂ₐ, ϕₐ)

Apply discrete differential operator against ϕₐ samples taken at k-simplicies.

apply(α, v)

Apply k-form α to v.

add boundary conditions to rhs of linear system

constrain( L, pin )

Add Diriclete boundary conditions to Linear Operator L. return tuple with reduced L and columns of L representing constrained dofs

interpolate( (a,b,c), edgevals, p)

Interpolate the discrete values on the edges of a triangle with points a,b,c to a vector at location p.

edgevals: values on edges should be ordered so the value at index i corresponds to the edge across from vertex i.

poisson(L, b, u_boundary, boundary_index)

Solve the Poisson problem Lu = b, where L is a Laplace-beltrami operator with Dirichlet boundary specified by uboundary at locations boundaryindex.

Δ(topo, P)

Discrete Laplace-Beltrami operator

local Laplace matrix for a triangle

Δcell(topo, P)

The symmetric Laplace-Beltrami operator. ⋆Δcell = Δ. The cell Δ differs from the (strong) Δ in that cell version maps to the dual simplex.

i.e. Δcell(topo, P) -> Ω⋆₀ and Δ(topo, P) -> Ω₀

The rhs must also be on dual. You must apply ⋆ to the rhs. Solve: Δ(topo, P) = b ––-> Solve: Δcell(topo, P) = ⋆(topo,P)*b

Useful for constructing symmetric operators.

Δcell(F, P)

Laplace-Beltrami operator built from list of triangle indices and points. Optimized for performance.

δ(topo, P)

Discrete coderivative

𝑑(curve)

Ω₀->𝑑₀->Ω₁ map from vertex to edges

⋆₀⁻¹->𝑑₀->Ω₁

Ω₀->𝑑₀->Ω₁

𝑑(topo, 𝑑₀ᵗ) differential of 𝑑₀ᵗ which is a map to dual vertices is zero, since dual simplex of degree 3 doesn't exist on 𝑅² manifold

𝑑(topo, 𝑑₁) differential of 𝑑₁ which is a map to faces is zero, since simplex of degree 3 doesn't exist on 𝑅² manifold

𝑑(topo, ⋆₂⁻¹) differential of ⋆₂⁻¹ which is a map to faces is zero, since simplex of degree 3 doesn't exist on 𝑅² manifold

𝑑(topo, ⋆₀) differential of ⋆₀ which is a map to dual vertices is zero, since dual simplex of degree 3 doesn't exist on 𝑅² manifold

𝑑( topo, s )

Ω₂⋆->𝑑₀->Ω⋆₁. A map to dual edges

𝑑( topo, Ω⋆₁ )

Ω⋆₁->𝑑₁->Ω⋆₂

𝑑₀->Ω⋆₁->𝑑₁->Ω⋆₂ ⋆₁->Ω⋆₁->𝑑₁->Ω⋆₂

𝑑( topo, Ω₁ )

 Ω₁->𝑑₁->Ω₂

𝑑₀->Ω₁->𝑑₁->Ω₂ ⋆⁻¹->Ω₁->𝑑₁->Ω₂

A map to 2-forms (faces)

𝑑(topo, Z) differential of zero is zero

differential of simple k-form

differential of simple 1-form

𝑑( k, 𝐼 )

differential of 0-form. 𝐼 is the psuedovector of the algebra

𝑑(k, jacobian)

Exterior Derivative of KVector of k-forms with current jacobian

𝑑(k)

Exterior derivative operator on KVector wedged with ZForms. Differential of k-form.

𝑑( k )

differential of 0-form

⋆( topo, Ω⋆₁ )

Ω⋆₁->⋆₁⁻¹->Ω₁

𝑑₀->Ω⋆₁->⋆₁⁻¹->Ω₁ ⋆₁->Ω⋆₁->⋆₁⁻¹->Ω₁

⋆(topo, P) Ω₀->⋆₀->Ω⋆₂. A map to dual 2-forms (dual vertices)

⋆(topo, P) Ω₀->⋆₀->Ω⋆₂. A map to dual 2-forms (dual vertices)

->⋆₂->⋆₂⁻¹->Ω₂

⋆(topo, P) Ω₀->⋆₀->Ω⋆₂. A map to dual 2-forms (dual vertices)

->⋆₀->⋆₀⁻¹->Ω₀

⋆( topo, Ω₁ )

 Ω₁->⋆₁->Ω⋆₁

𝑑₀->Ω₁->⋆₁->Ω⋆₁ ⋆⁻₁->Ω₁->⋆₁->Ω⋆₁

A map to 1-forms (cell edges)

⋆( topo, Ω₂ )

 Ω₂->⋆₂->Ω⋆₀

𝑑₁->Ω₂->⋆₂->Ω⋆₀ ⋆⁻₂->Ω₂->⋆₂->Ω⋆₀

A map to 0-forms (cell centred vertex)

    Ω₀->⋆₀->Ω⋆₂

⋆₀⁻¹->Ω₀->⋆₀->Ω⋆₂

A map to 2-forms (vertex cell).

across( topo, P, heh )

the triangle vertex across from the halfedge.

the cotangent of the angle between two vectors

cotanweight( topo, P, h )

calculate cotangent weight across an edge. 0.5*(cot(α) + cot(Β)), where α,Β are the angles of edges opposite our halfedge

vector cotangent of angle at Pₐ, ccw from P₁ to P₂

Barycentric dual vertex area. One third of the summed areas of triangles incident to given vertex