Filling a 3D-Body with a systematic point raster - r

Question

I have a set of 3D-Bodies. Each Body is defined by 8 points with three coordinates each. All of the bodies are cubical or approximately cubical. I would like to "fill" the cubes with a systematic point raster. The coordinates are stored in simple data.frames.
I developed the following code that does what I want for cubical bodies:
# libraries
library(rgl)
# define example cube with 8 points
excube <- data.frame(
x = c(1,1,1,1,5,5,5,5),
y = c(1,1,4,4,1,1,4,4),
z = c(4,8,4,8,4,8,4,8)
)
# cubeconst: fill cube (defined by 8 corner points) with a 3D-point-raster
cubeconst <- function(x, y, z, res) {
cube <- data.frame()
xvec = seq(min(x), max(x), res)
yvec = seq(min(y), max(y), res)
zvec = seq(min(z), max(z), res)
for (xpoint in 1:length(xvec)) {
for (ypoint in 1:length(yvec)) {
for (zpoint in 1:length(zvec)) {
cube <- rbind(cube, c(xvec[xpoint], yvec[ypoint], zvec[zpoint]))
}
}
}
colnames(cube) <- c("x", "y", "z")
return(cube)
}
# apply cubeconst to excube
fcube <- cubeconst(x = excube$x, y = excube$y, z = excube$z, res = 0.5)
# plot result
plot3d(
fcube$x,
fcube$y,
fcube$z,
type = "p",
xlab = "x",
ylab = "y",
zlab = "z"
)
Now I'm searching for a solution to "fill" approximately cubical bodies like for example the following body:
# badcube
badcube <- data.frame(
x = c(1,1,1,1,5,5,5,5),
y = c(1,1,4,4,1,1,4,4),
z = c(4,10,4,12,4,8,4,8)
)
# plot badcube
plot3d(
badcube$x,
badcube$y,
badcube$z,
col = "red",
size = 10,
type = "p",
xlab = "x",
ylab = "y",
zlab = "z"
)
Maybe you can point me in the right direction.

Answer 1

You need to transform the hexahedron (wonky cube) to a unit cube. The following image shows what I mean, and gives us a numbering scheme for the vertices of the hexa. Vertex 2 is hidden behind the cube.
The transformation is from real space x,y,z, to a new coordinate system u,v,w, in which the hexa is a unit cube. The typical function used for hexa looks like this.
x = A + B*u + C*v + D*w + E*u*v + F*u*w + G*v*w + H*u*v*w
Transformations for y and z coordinates are of the same form. You have 8 corners to your cube, so you can substitute these in to solve for coefficients A,B,.... The unit coordinates u,v,w are either 0 or 1 at every vertex, so this simplifies things a lot.
x0 = A // everything = zero
x1 = A + B // u = 1, others = zero
x2 = A + C // v = 1, ...
x4 = A + D // w = 1
x3 = A + B + C + E // u = v = 1
x5 = A + B + D + F // u = w = 1
x6 = A + C + D + G // v = w = 1
x7 = A + B + C + D + E + F + G + H // everything = 1
You then have to solve for A,B,.... This is easy because you just forward substitute. A equals x0. B equals x1 - A, etc... You have to do this for y and z also, but if your language supports vector operations, this can probably be done in the same step as for x.
Once you have the coefficients, you can convert a point u,v,w to x,y,z. Now, if you have a point generation scheme which works on a 1x1x1 cube, you can transform the result to the origonal hex. You could retain the same triple-loop structure in your posted code, and vary u,v,w between 0 and 1 to create a grid of points within the hex.
I'm afraid I don't know r, so I can't give you any example code in that language. Here's a quick python3 example, though, just to prove it works.
import matplotlib.pyplot as pp
import numpy as np
from mpl_toolkits.mplot3d import Axes3D
np.random.seed(0)
cube = np.array([
[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [1.0, 1.0, 0.0],
[0.0, 0.0, 1.0], [1.0, 0.0, 1.0], [0.0, 1.0, 1.0], [1.0, 1.0, 1.0]])
hexa = cube + 0.5*np.random.random(cube.shape)
edges = np.array([
[0, 1], [0, 2], [1, 3], [2, 3],
[0, 4], [1, 5], [2, 6], [3, 7],
[4, 5], [4, 6], [5, 7], [6, 7]])
def cubeToHexa(hexa, u, v, w):
A = hexa[0]
B = hexa[1] - A
C = hexa[2] - A
D = hexa[4] - A
E = hexa[3] - A - B - C
F = hexa[5] - A - B - D
G = hexa[6] - A - C - D
H = hexa[7] - A - B - C - D - E - F - G
xyz = (
A +
B*u[...,np.newaxis] +
C*v[...,np.newaxis] +
D*w[...,np.newaxis] +
E*u[...,np.newaxis]*v[...,np.newaxis] +
F*u[...,np.newaxis]*w[...,np.newaxis] +
G*v[...,np.newaxis]*w[...,np.newaxis] +
H*u[...,np.newaxis]*v[...,np.newaxis]*w[...,np.newaxis])
return xyz[...,0], xyz[...,1], xyz[...,2]
fg = pp.figure()
ax = fg.add_subplot(111, projection='3d')
temp = np.reshape(np.append(hexa[edges], np.nan*np.ones((12,1,3)), axis=1), (36,3))
ax.plot(temp[:,0], temp[:,1], temp[:,2], 'o-')
u, v, w = np.meshgrid(*[np.linspace(0, 1, 6)]*3)
x, y, z = cubeToHexa(hexa, u, v, w)
ax.plot(x.flatten(), y.flatten(), z.flatten(), 'o')
pp.show()
I can't recall the exact justificiation for this form of the transformation. It's certainly easy to solve, and it has no squared terms, so lines in the directions of the u,v,w axes map to straight lines in x,y,z. This means your cube edges and faces are guaranteed to conform, as well as the corners. I lack the maths to prove it, though, and I couldn't find any googleable information either. My knowledge comes from a distant memory of textbooks on Finite Element Methods, where these sort of transformations are common. If you need more information, I suggest you start looking there.

Answer 2

Thanks to Bills explanation and examples I was able to come up with the following solution in R:
# libraries
library(rgl)
# create heavily distorted cube - hexahedron
hexatest <- data.frame(
x = c(0,1,0,4,5,5,5,5),
y = c(1,1,4,4,1,1,4,4),
z = c(4,8,4,9,4,8,4,6)
)
# cubetohexa: Fills hexahedrons with a systematic point raster
cubetohexa <- function(hexa, res){
# create new coordinate system (u,v,w)
resvec <- seq(0, 1, res)
lres <- length(resvec)
u <- c()
for (p1 in 1:lres) {
u2 <- c()
for (p2 in 1:lres) {
u2 <- c(u2, rep(resvec[p2], lres))
}
u <- c(u,u2)
}
v <- c()
for (p1 in 1:lres) {
v <- c(v, rep(resvec[p1], lres^2))
}
w <- rep(resvec, lres^2)
# transformation
A <- as.numeric(hexa[1,])
B <- as.numeric(hexa[2,]) - A
C <- as.numeric(hexa[3,]) - A
D <- as.numeric(hexa[5,]) - A
E <- as.numeric(hexa[4,]) - A - B - C
F <- as.numeric(hexa[6,]) - A - B - D
G <- as.numeric(hexa[7,]) - A - C - D
H <- as.numeric(hexa[8,]) - A - B - C - D - E - F - G
A <- matrix(A, ncol = 3, nrow = lres^3, byrow = TRUE)
B <- matrix(B, ncol = 3, nrow = lres^3, byrow = TRUE)
C <- matrix(C, ncol = 3, nrow = lres^3, byrow = TRUE)
D <- matrix(D, ncol = 3, nrow = lres^3, byrow = TRUE)
E <- matrix(E, ncol = 3, nrow = lres^3, byrow = TRUE)
F <- matrix(F, ncol = 3, nrow = lres^3, byrow = TRUE)
G <- matrix(G, ncol = 3, nrow = lres^3, byrow = TRUE)
H <- matrix(H, ncol = 3, nrow = lres^3, byrow = TRUE)
for (i in 1:(lres^3)) {
B[i,] <- B[i,] * u[i]
C[i,] <- C[i,] * v[i]
D[i,] <- D[i,] * w[i]
E[i,] <- E[i,] * u[i] * v[i]
F[i,] <- F[i,] * u[i] * w[i]
G[i,] <- G[i,] * v[i] * w[i]
H[i,] <- H[i,] * u[i] * v[i] * w[i]
}
m <- data.frame(A+B+C+D+E+F+G+H)
colnames(m) <- c("x", "y", "z")
# output
return(m)
}
# apply cubetohexa to hexatest
cx <- cubetohexa(hexatest, 0.1)
# plot result
plot3d(
cx$x,
cx$y,
cx$z,
type = "p",
xlab = "x",
ylab = "y",
zlab = "z"
)
Edit:
This function is now implemented with Rcpp in my R package recexcavAAR.