Not an R question directly, but has anyone got a method for placing a moderately large number of (near) equi-spaced points on a sphere? I have a nasty feeling platonic solids are needed for exact solutions and I'm thinking of samplings involving around 200 - 1000 regularly-spaced points, Thanks, Richard Rowe Richard Rowe Senior Lecturer Department of Zoology and Tropical Ecology, James Cook University Townsville, Queensland 4811, Australia fax (61)7 47 25 1570 phone (61)7 47 81 4851 e-mail: Richard.Rowe at jcu.edu.au http://www.jcu.edu.au/school/tbiol/zoology/homepage.html -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- r-help mailing list -- Read http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html Send "info", "help", or "[un]subscribe" (in the "body", not the subject !) To: r-help-request at stat.math.ethz.ch _._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._
This puts n points in R^p "as equally spaced as possible". If
sphere=TRUE
it constrains them to be on the sphere.
===========================================================
equidist<-function(n, p, eps=1e-6, itmax=Inf, verbose=TRUE,
sphere=FALSE) {
x<-matrix(rnorm(n*p),n,p); sprev<-0; itel<-1
if (sphere) x<-row.norm(x) else x<-x/sqrt(sum(x^2))
repeat{
s<-sum(d<-eudist(x))
if (verbose) cat(formatC(itel,width=6),
formatC(s,digits=6,width=20,format="f"),"\n")
if ((((s-sprev)/sprev) < eps) || (itel == itmax)) return(x)
diag(d)<-1; e<-1/d; f<-apply(e,1,sum); e<--e; diag(e)<-f
x<-e%*%x
if (sphere) x<-row.norm(x) else x<-x/sqrt(sum(x^2))
sprev<-s; itel<-itel+1
}
}
row.norm<-function(x) x/sqrt(apply(x^2,1,sum))
eudist<-function(x){
c<-crossprod(t(x)); s<-diag(c)
sqrt(outer(s,s,"+")-2*c)
}
bmat<-function(d,delta,w){
}
On Friday, October 25, 2002, at 04:46 PM, Richard Rowe wrote:
> Not an R question directly, but has anyone got a method for placing a
> moderately large number of (near) equi-spaced points on a sphere? I
> have a nasty feeling platonic solids are needed for exact solutions
> and I'm thinking of samplings involving around 200 - 1000
> regularly-spaced points,
>
> Thanks,
>
> Richard Rowe
>
> Richard Rowe
> Senior Lecturer
> Department of Zoology and Tropical Ecology, James Cook University
> Townsville, Queensland 4811, Australia
> fax (61)7 47 25 1570
> phone (61)7 47 81 4851
> e-mail: Richard.Rowe at jcu.edu.au
> http://www.jcu.edu.au/school/tbiol/zoology/homepage.html
>
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>
==Jan de Leeuw; Professor and Chair, UCLA Department of Statistics;
Editor: Journal of Multivariate Analysis, Journal of Statistical
Software
US mail: 9432 Boelter Hall, Box 951554, Los Angeles, CA 90095-1554
phone (310)-825-9550; fax (310)-206-5658; email: deleeuw at stat.ucla.edu
homepage: http://gifi.stat.ucla.edu
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On Fri, 2002-10-25 at 17:46, Richard Rowe wrote:> Not an R question directly, but has anyone got a method for placing a > moderately large number of (near) equi-spaced points on a sphere? I have a > nasty feeling platonic solids are needed for exact solutions and I'm > thinking of samplings involving around 200 - 1000 regularly-spaced points,There's some C source for that here: http://astronomy.swin.edu.au/~pbourke/geometry/spherepoints/ Regards, Luke Hutchison. -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- r-help mailing list -- Read http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html Send "info", "help", or "[un]subscribe" (in the "body", not the subject !) To: r-help-request at stat.math.ethz.ch _._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._
I don't have code for this in R, but a general technique I've used is as follows: Start with 4 points at the corners of a regular tetrahedron. Then recursively subdivide each triangular face into 3 triangular faces by placing a point at the center of it. This is easy, just average the 3 vertices and normalize to length 1. The only tricky part is keeping track of which vertices form faces. Duncan Murdoch -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- r-help mailing list -- Read http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html Send "info", "help", or "[un]subscribe" (in the "body", not the subject !) To: r-help-request at stat.math.ethz.ch _._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._
Hi Richard,
Here's a non-iterative way of approximately laying out points on the surface
of a sphere. It basically lays out a spiral on the surface (like peeling an
apple) then drops the points along it. I don't know if it will be any actual
use to you, but I quite like it (and would be interested to know what breaks
it).
Cheers,
Mike.
----------------------------------------------------------------
Mike Lonergan
The Observatory, Buchanan Gardens ph: (+44) 1334 461803
University of St Andrews fax:(+44) 1334 461800
Fife, KY16 9LZ, Scotland email: mel at mcs.st-and.ac.uk
--------------------------------------------------------------------
##########
points.on.unit.sphere<-function(n)
{ # attempts to regularly distribute approximately n points on
# the surface of a unit sphere non-iteratively
# by laying them out along a spiral with a fixed (angular) pitch, c,
# x,y,z are the cartesian coordinates of the points,
# theta is their longitude, phi their lattitude (in radians)
# by Mike Lonergan, mel at mcs.st-and.ac.uk
c<-sqrt(n*pi)/2
theta<-c(0,2*pi)
for(i in 3:floor(n/2))
theta[i]<-theta[i-1]+pi/(c*cos(theta[i-1]/(2*c)-pi/2))
# theta[i]<-sqrt(2*c+theta[i-1]^2)
if (2*floor(n/2)==n)
theta<-c(theta,2*pi*c-rev(theta))
else
theta<-c(theta, pi*c, 2*pi*c-rev(theta))
pts<-data.frame(theta=theta)
pts$phi<-theta/(2*c)-pi/2
pts$x<-cos(theta)*cos(pts$phi)
pts$y<-sin(theta)*cos(pts$phi)
pts$z<-sin(pts$phi)
pts
}
nearest<-function(data)
{ # takes a dataframe with columns x, y, z and returns the (straightline)
nearest
# neighbour distances between the points in its rows
# inefficient, but adequate for checking points.on.unit.sphere.
res<-NA
for (i in 1:dim(data)[1])
res[i]<-sqrt(min((data$x[-i]-data$x[i])^2 + (data$y[-i]-data$y[i])^2 +
(data$z[-i]-data$z[i])^2))
res
}
points.on.unit.sphere(1000)->pous
nearest(pous)->npous
par(mfrow=c(2,2))
hist(npous,main="nearest neighbour distances")
plot(pous$x+sign(pous$z),pous$y,main="plan view")
plot(pous$x+sign(pous$y),pous$z,main="side.view")
plot(npous,ylab="nearest neighbour distances",xlab="theta")
length(which(npous<0.06))
##########
> -----Original Message-----
> From: owner-r-help at stat.math.ethz.ch
> [mailto:owner-r-help at stat.math.ethz.ch]On Behalf Of Richard Rowe
> Sent: 26 October 2002 00:46
> To: r-help at stat.math.ethz.ch
> Subject: [R] points on a sphere
>
>
> Not an R question directly, but has anyone got a method for
> placing a
> moderately large number of (near) equi-spaced points on a
> sphere? I have a
> nasty feeling platonic solids are needed for exact solutions and
I'm
> thinking of samplings involving around 200 - 1000
> regularly-spaced points,
>
> Thanks,
>
> Richard Rowe
>
> Richard Rowe
> Senior Lecturer
> Department of Zoology and Tropical Ecology, James Cook University
> Townsville, Queensland 4811, Australia
> fax (61)7 47 25 1570
> phone (61)7 47 81 4851
> e-mail: Richard.Rowe at jcu.edu.au
> http://www.jcu.edu.au/school/tbiol/zoology/homepage.html
>
> -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
> .-.-.-.-.-.-.-.-.-
> r-help mailing list -- Read
> http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html
> Send "info", "help", or "[un]subscribe"
> (in the "body", not the subject !) To:
> r-help-request at stat.math.ethz.ch
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