Creates a multi-resolution grid based on subdividing triangles starting with faces of an icosahedron.
IcosahedronGrid(K)IcosahedronFaces(K)
IcosahedronGrid A list with K components each component is a three column matrix
giving the direction cosines for each grid point.
IcosahedronFaces A list with components:
The same list returned by IcosahedronGrid.
A list with K-1 components. Each components are the faces at a given level
represented as a three dimensional array (3X3XN with N the number of faces and a given level). The
array indices are vertices of triangle, coordinates and faces within a resolution level. e.g. to
extract the 10th face (out of 80) for the 2nd level:
look<- IcosahedronFaces(3)$Faces
triangle <- (look[[2]])[,,10]
print (triangle)
[,1] [,2] [,3]
[1,] -0.5257311 -0.8506508 0.000000
[2,] -0.8090170 -0.5000000 0.309017
[3,] -0.8090170 -0.5000000 -0.309017
rowSums( triangle^2)
[1] 1 1 1
triangle will be the 10th face for the second level where
the columns are the 3d
coordinates of the direction cosines and the rows index the three
vertices.
Number of levels.
Doug Nychka and Zachary Thomas
Creates a nearly regular grid by taking the first level as the 12 points from a regular icosahedron. The subsequent levels generate a finer set of points by subdividing each triangular face into 4 new triangles. The three new mid points from the subdivision are added to the previous nodes to give the new level of resolution. The triangles tend to be roughly equilateral and so the nodes will tend to be roughly equally space but there is some variation in distances among nearest neighbors.
To depict the faces and nodes in a snazzy way use the rgl package and the following code
library( rgl)
# show level 3
Level<- 3
SGrid <- IcosahedronFaces(4)
Tri <- SGrid$Face[[Level]]
L <- dim(Tri)[3]
plot3d(rbind(c(0, 0, 0)), xlim = c(-1, 1), ylim = c(-1, 1),
zlim = c(-1, 1), axes = FALSE, xlab = "", ylab = "",
zlab = "", type = "p", box = TRUE)
for (k in 1:L) {
U <- Tri[, , k]
rgl.triangles(U, col = "grey80")
}
plot3d(SGrid$nodes[[Level]], col = "green4", type = "s",
radius = 0.03, add = TRUE)
toSphere, directionCosines, projectionSphere
# second level in lon lat coordinates
look<- IcosahedronGrid(3)
lonlat<- toSphere( look[[3]])
plot( lonlat, xlab="lon", ylab="lat")
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