R: Define a mesh on the unit sphere/ball in n-dimensions...
PolarSphere
R Documentation
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 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: