frameSphere {frechet}

Generate a "natural" frame (orthonormal basis)

Description

Generate a "natural" frame (orthonormal basis) for the tangent space at x on the unit sphere.

Usage

frameSphere(x)

Arguments

x

A unit vector of length d.

Details

The first (i+1) elements of the ith basis vector are given by \sin\theta_i\prod_{j=1}^{i-1}\cos\theta_j, \sin\theta_i\sin\theta_1 \prod_{j=2}^{i-1}\cos\theta_j, \sin\theta_i\sin\theta_2 \prod_{j=3}^{i-1}\cos\theta_j, \dots, \sin\theta_i\sin\theta_{i-1}, -\cos\theta_i, respectively. The rest elements (if any) of the ith basis vector are all zero.

Value

A d-by-(d-1) matrix where columns hold the orthonormal basis of the tangent space at x on the unit sphere.

Examples

frameSphere(c(1,0,0,0))