Polar2Rec {KEPTED}

Polar to rectangular coordinates

Description

Given a polar coordinate representation (R,Theta) of a d-dimensional vector X, where R is the length of X and the (d-1)-dimensional vector Theta contains the d-1 angles of X, this function compute X in its rectangular coordinate representation.

Usage

Polar2Rec(R, Theta)

Arguments

R

The length of X.

Theta

A vector of length d-1, containing the angles of X.

Details

The formula corresponds to v=rho(theta) as in Lemma 1 of Tang and Li (2024). See also Anderson (2003). Note that when d=2, V will be (sin(Theta),cos(Theta)).

Value

A list of the following:

X

A vector in rectangular coordinate.

V

The directional vector of X. Note that V is always on the unit sphere.

References

Tang, Y. and Li, B. (2024), “A nonparametric test for elliptical distribution based on kernel embedding of probabilities,” https://arxiv.org/abs/2306.10594 Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis. John Wiley & Suns, Inc. Huboken, New Jersey.

Examples

R=2
Theta=c(pi/6,pi/3)
Polar2Rec(R,Theta)