basisTan.Euclidean {manifold}

Obtain an orthonormal basis on the tangent space

Description

Parametrize the tangent space at location p, so that the parameterized version contains an open neighborhood around the origin. (The dimension of v is potentially reduced).

Usage

## S3 method for class 'Euclidean'
basisTan(mfd, p)

basisTan(mfd, p)

## S3 method for class 'SO'
basisTan(mfd, p)

## S3 method for class 'SPD'
basisTan(mfd, p)

## S3 method for class 'Sphere'
basisTan(mfd, p)

## S3 method for class 'FlatTorus'
basisTan(mfd, p)

Arguments

mfd

A manifold object created by createM

p

A vector for a base point on the manifold

Value

An orthonormal basis matrix D, whose columns contain the basis vectors, so that 'D^T v' give the coordinates 'v0' for a tangent vector 'v', and 'D

Methods (by class)

• basisTan(Euclidean): An identity matrix

• basisTan(SO): An identity matrix

• basisTan(SPD): The basis is obtained from enumerating the (non-strict) lower-triangle of a square matrix. If i != j, the (i, j)th entry is mapped into a matrix with 1/sqrt(2) in the (i,j) and (j,i) entries and 0 in other entries; if (i == j), it is mapped to a matrix with 1 in the ith diagonal element and 0 otherwise. The mapped matrix is then vectorized to obtain the basis vector.

• basisTan(Sphere): The basis at the north pole is [0, ..., 1, ..., 0] where the 1 is at the j = 2, ..., dAmbth location. The basis at a point p is obtained through rotating the basis from the north pole to p along the shortest geodesic.

• basisTan(FlatTorus): An identity matrix