DiscreteDifferentialGeometry.CellKForm โ Type
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 )

DiscreteDifferentialGeometry.PrimalKForm โ Type
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

DiscreteDifferentialGeometry.apply โ Method
apply(๐ฬโ, ฮฑ, P, k)

Apply discrete differential operator ๐ฬโ to k-form ฮฑ at points P with k-vector k.

DiscreteDifferentialGeometry.apply โ Method
apply(๐ฬโ, ฯโ)

Apply discrete differential operator against ฯโ samples taken at k-simplicies.

DiscreteDifferentialGeometry.interpolate โ Method
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.

DiscreteDifferentialGeometry.poisson โ Method
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.

DiscreteDifferentialGeometry.ฮcell โ Method
ฮcell(topo, P)

The symmetric Laplace-Beltrami operator. โฮcell = ฮ. The cell ฮ differs from the (strong) ฮ in that cell version maps to the dual simplex.

i.e. ฮcell(topo, P) -> ฮฉโโ and ฮ(topo, P) -> ฮฉโ

The rhs must also be on dual. You must apply โ to the rhs. Solve: ฮ(topo, P) = b โโ-> Solve: ฮcell(topo, P) = โ(topo,P)*b

Useful for constructing symmetric operators.

DiscreteDifferentialGeometry.๐ โ Method

๐(topo, ๐โแต) differential of ๐โแต which is a map to dual vertices is zero, since dual simplex of degree 3 doesn't exist on ๐ยฒ manifold

DiscreteDifferentialGeometry.๐ โ Method

๐(topo, ๐โ) differential of ๐โ which is a map to faces is zero, since simplex of degree 3 doesn't exist on ๐ยฒ manifold

DiscreteDifferentialGeometry.๐ โ Method

๐(topo, โโโปยน) differential of โโโปยน which is a map to faces is zero, since simplex of degree 3 doesn't exist on ๐ยฒ manifold

DiscreteDifferentialGeometry.๐ โ Method

๐(topo, โโ) differential of โโ which is a map to dual vertices is zero, since dual simplex of degree 3 doesn't exist on ๐ยฒ manifold

DiscreteDifferentialGeometry.๐ โ Method
๐( topo, ฮฉโโ )

ฮฉโโ->๐โ->ฮฉโโ

๐โ->ฮฉโโ->๐โ->ฮฉโโ โโ->ฮฉโโ->๐โ->ฮฉโโ

DiscreteDifferentialGeometry.๐ โ Method
๐( topo, ฮฉโ )

ฮฉโ->๐โ->ฮฉโ

๐โ->ฮฉโ->๐โ->ฮฉโ โโปยน->ฮฉโ->๐โ->ฮฉโ

A map to 2-forms (faces)

Multivectors.:โ โ Method
โ( topo, ฮฉโโ )

ฮฉโโ->โโโปยน->ฮฉโ

๐โ->ฮฉโโ->โโโปยน->ฮฉโ โโ->ฮฉโโ->โโโปยน->ฮฉโ

Multivectors.:โ โ Method

โ(topo, P) ฮฉโ->โโ->ฮฉโโ. A map to dual 2-forms (dual vertices)

Multivectors.:โ โ Method

โ(topo, P) ฮฉโ->โโ->ฮฉโโ. A map to dual 2-forms (dual vertices)

Multivectors.:โ โ Method

โ(topo, P) ฮฉโ->โโ->ฮฉโโ. A map to dual 2-forms (dual vertices)

Multivectors.:โ โ Method
โ( topo, ฮฉโ )

ฮฉโ->โโ->ฮฉโโ

๐โ->ฮฉโ->โโ->ฮฉโโ โโปโ->ฮฉโ->โโ->ฮฉโโ

A map to 1-forms (cell edges)

Multivectors.:โ โ Method
โ( topo, ฮฉโ )

ฮฉโ->โโ->ฮฉโโ

๐โ->ฮฉโ->โโ->ฮฉโโ โโปโ->ฮฉโ->โโ->ฮฉโโ

A map to 0-forms (cell centred vertex)

Multivectors.:โ โ Method
ฮฉโ->โโ->ฮฉโโ

โโโปยน->ฮฉโ->โโ->ฮฉโโ

A map to 2-forms (vertex cell).

DiscreteDifferentialGeometry.cotanweight โ Method

cotanweight( topo, P, h )

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