mvmesh (version 1.2)

PolarSphere: Define a mesh on the unit sphere/ball in n-dimensions determined by a polar coordinates grid.

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

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 RectangularMesh.

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.

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

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.

Examples

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 )

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) ) )