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frechet (version 0.3.0)

frameSphere: 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)

Value

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

Arguments

x

A unit vector of length \(d\).

Details

The first \((i+1)\) elements of the \(i\)th 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 \(i\)th basis vector are all zero.

Examples

Run this code
frameSphere(c(1,0,0,0))

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