# PolarSphere: Define a mesh on the unit sphere/ball in n-dimensions... In mvmesh: Multivariate Meshes and Histograms in Arbitrary Dimensions

## Description

Subdivide the unit ball or sphere into simplices in arbitrary dimensions using a rectangular grid on the polar parameterization of the sphere.

The general n-dimensional polar coordinates to and from rectangular coordinates transformations are provided.

## Usage

 ```1 2 3 4``` ```PolarSphere(n, breaks=c(rep(4,n-2),8), p = 2, positive.only = FALSE) PolarBall( n, breaks=c(rep(4,n-2),8), p=2, positive.only=FALSE ) Rectangular2Polar( x ) Polar2Rectangular( r, theta ) ```

## Arguments

 `n` Dimension of the space; the Polar sphere is an (n-1) dimensional manifold `breaks` specification of the partition of in the angle space theta. See the definition of 'breaks' in `SolidRectangle`. `p` Power used in the l^p norm; p=2 is the Euclidean norm `positive.only` TRUE means restrict to the positive orthant; FALSE gives the full ball `r` a vector of radii of length m. `theta` a (n-1) x m matrix of angles. `x` (n x m) matrix, with column j being the point in n-dimensional space.

## Details

`PolarSphere` computes an approximation to the unit sphere using a rectangular grid in the polar angle space. `PolarBall` uses a partition of the polar sphere and joins those simplices to the origin to approximately partition the unit ball. `LpNorm` computes the l^p norm of each columns of `x`.

`Polar2Rectangular` and `Rectangular2Polar` convert between the polar coordinate representation (r,theta[1],...,theta[n-1]) and the rectangular coordinates (x[1],...,x[n]).

n dimensional polar coordinates are given by the following:
rectangular x=(x[1],...,x[n]) corresponds to polar (r,theta[1],...,theta[n-1]) by
x[1] = r*cos(theta[1])
x[2] = r*sin(theta[1])*cos(theta[2])
x[3] = r*sin(theta[1])*sin(theta[2])*cos(theta[3])
...
x[n-1]= r*sin(theta[1])*sin(theta[2])*...*sin(theta[n-2])*cos(theta[n-1])
x[n] = r*sin(theta[1])*sin(theta[2])*...*sin(theta[n-2])*sin(theta[n-1])

Here theta[1],...,theta[n-2] in [0,pi), and theta[n-1] in [0,2*pi). This is the parameterization described in the Wikipedia webpage for "n-sphere". Note that this is NOT a 1-1 transformation: when theta[1]=0, it follows that x[2]=x[3]=...=x[n]=0. This is analagous to all longitude lines going through the north pole in standard 3d spherical coordinates.

For multivariate integration, the Jacobian of the above tranformation is J(theta) = r^(n-1) * prod( sin(theta[1:(n-2)])^((n-2):1) ); note that theta[n-1] does not appear in the Jacobian.

## Value

`PolarSphere` and `PolarBall` return an object of class "mvmesh" as described in `mvmesh`. `Polar2Rectangular` returns an (n x m) matrix of rectangular coordinates. `Rectangular2Polar` returns a list with fields:

 `r` a vector of length m containing the radii `theta` an (n x m) matrix of angles

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19``` ```PolarSphere( n=3, breaks=4) PolarBall( n=3, breaks=4 ) (x <- matrix( 1:10, ncol=2 )) (a <- Rectangular2Polar( x )) Polar2Rectangular( a\$r, a\$theta ) (x <- matrix( 1:12, ncol=4 )) (a <- Rectangular2Polar( x )) Polar2Rectangular( a\$r, a\$theta ) ## Not run: plot( PolarSphere( n=2, breaks=8 ) ) plot( PolarBall( n=2, breaks=8 ) ) plot( PolarSphere( n=3, breaks=c(4,8) ) ) plot( PolarBall( n=3, breaks=c(4,8) ) ) ## End(Not run) ```

mvmesh documentation built on Feb. 12, 2020, 1:09 a.m.