Discrete Exterior Calculus

Discrete exterior calculus. More...

Functions
template<typename Mesh >
int	Euclid::halfedge_orientation (const Mesh &mesh, halfedge_t< Mesh > h)
	Give a consistent orientation for halfedges. More...
template<typename Mesh , typename T >
void	Euclid::d0 (const Mesh &mesh, Eigen::SparseMatrix< T > &op)
	Exterior derivative operator of 0-form. More...
template<typename Mesh , typename T >
void	Euclid::d1 (const Mesh &mesh, Eigen::SparseMatrix< T > &op)
	Exterior derivative operator of 1-form. More...
template<typename Mesh , typename T >
void	Euclid::star1 (const Mesh &mesh, Eigen::SparseMatrix< T > &op)
	Hodge star operator of 1-form. More...
template<typename Mesh , typename DerivedA , typename DerivedB , typename DerivedC >
void	Euclid::wedge1 (const Mesh &mesh, const Eigen::MatrixBase< DerivedA > &w1, const Eigen::MatrixBase< DerivedB > &w2, Eigen::MatrixBase< DerivedC > &result)
	The primal x primal wedge product. More...
template<typename Mesh , typename DerivedA , typename DerivedB , typename DerivedC >
void	Euclid::star_wedge1 (const Mesh &mesh, const Eigen::MatrixBase< DerivedA > &w1, const Eigen::MatrixBase< DerivedB > &w2, Eigen::MatrixBase< DerivedC > &result)
	The primal x dual wedge product. More...

Detailed Description

This package provides common DEC operators.

Reference

[1] Elcott, S., Schroder, P. Building Your Own DEC at Home.

[2] Gu, X., Yau, S.T. Global Conformal Surface Parameterization.

Function Documentation

template<typename Mesh , typename T >

void Euclid::d0	(	const Mesh &	mesh,
		Eigen::SparseMatrix< T > &	op
	)

Exterior derivative for 0-form f is \(D0 * f\).

Parameters

mesh	Target triangle mesh.
op	An E x V matrix acting as the exterior derivative operator.

template<typename Mesh , typename T >

void Euclid::d1	(	const Mesh &	mesh,
		Eigen::SparseMatrix< T > &	op
	)

Exterior derivative for 1-form w is \(D1 * w\).

Parameters

mesh	Target triangle mesh.
op	An F x E matrix acting as the exterior derivative operator.

template<typename Mesh >

int Euclid::halfedge_orientation	(	const Mesh &	mesh,
		halfedge_t< Mesh >	h
	)

Parameters

mesh	Target triangle mesh.
h	Target halfedge.

template<typename Mesh , typename T >

void Euclid::star1	(	const Mesh &	mesh,
		Eigen::SparseMatrix< T > &	op
	)

Equivalent to the cotangent matrix.

Parameters

mesh	Target triangle mesh.
op	An E x E matrix acting as the hodge star operator.

template<typename Mesh , typename DerivedA , typename DerivedB , typename DerivedC >

void Euclid::star_wedge1	(	const Mesh &	mesh,
		const Eigen::MatrixBase< DerivedA > &	w1,
		const Eigen::MatrixBase< DerivedB > &	w2,
		Eigen::MatrixBase< DerivedC > &	result
	)

The wedge product is not linear so it cannot be represented as a matrix. Instead, we take the 1-forms as input and return the result.

Parameters

mesh	Target triangle mesh.
w1	1-form.
w2	1-form.
result	An F x 1 matrix storing the result of wedge product.

template<typename Mesh , typename DerivedA , typename DerivedB , typename DerivedC >

void Euclid::wedge1	(	const Mesh &	mesh,
		const Eigen::MatrixBase< DerivedA > &	w1,
		const Eigen::MatrixBase< DerivedB > &	w2,
		Eigen::MatrixBase< DerivedC > &	result
	)

The wedge product is not linear so it cannot be represented as a matrix. Instead, we take the 1-forms as input and return the result.

Parameters

mesh	Target triangle mesh.
w1	1-form.
w2	1-form.
result	An F x 1 matrix storing the result of wedge product.