Description Usage Arguments Details Value Examples
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.
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 )
|
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 |
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. |
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.
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 |
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)
|
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.