The compound of five tetrahedra
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In this previous post I was wrong when I said I didn’t have a tool to compute the intersection of the compound of five tetrahedra:
This can be achieved with the help of the rcdd package in R.
This package is a wrapper of the C library cddlib. It implements the double description of convex polyhedra:

the Vdescription is the description by the vertices of the polyhedron;

the Hdescription is the description by a set of linear inequalities.
Once we get the Hdescription of two or more polyhedra, it is straightforward to get the Hdescription of their intersection: it suffices to join all the linear inequalities. Then, cddlib will give us the corresponding Vrepresentation. That’s it.
The cddlib library is also usable in Python, with the pycddlib library. I’m going to show how to compute the intersection of the five tetrahedra forming the compound with R and with Python.
The R way
Here are the vertices of the five tetrahedra:
# the twenty vertices #### phi < (1 + sqrt(5)) / 2 a < 1 / sqrt(3) b < a / phi c < a * phi vertices < rbind( c( a, a, a), c( a, a, a), c( a, a, a), c(a, a, a), c(a, a, a), c(a, a, a), c( 0, b, c), c( 0, b, c), c( 0, b, c), c( c, 0, b), c(c, 0, b), c(c, 0, b), c( b, c, 0), c( b, c, 0), c(b, c, 0), c(b, c, 0), c( 0, b, c), c( a, a, a), c( c, 0, b), c(a, a, a) ) # the five tetrahedra #### th1 < vertices[c(17L, 14L, 2L, 11L), ] th2 < vertices[c(18L, 1L, 4L, 5L), ] th3 < vertices[c(19L, 6L, 15L, 7L), ] th4 < vertices[c( 3L, 13L, 12L, 8L), ] th5 < vertices[c(20L, 16L, 10L, 9L), ]
To make a Vdescription with rcdd, one uses the
function makeV
:
library(rcdd) V1 < makeV(th1) V2 < makeV(th2) V3 < makeV(th3) V4 < makeV(th4) V5 < makeV(th5)
Let’s look at a Vdescription in R:
V1 ## [,1] [,2] [,3] [,4] [,5] ## [1,] 0 1 0.0000000 0.3568221 0.9341724 ## [2,] 0 1 0.3568221 0.9341724 0.0000000 ## [3,] 0 1 0.5773503 0.5773503 0.5773503 ## [4,] 0 1 0.9341724 0.0000000 0.3568221 ## attr(,"representation") ## [1] "V"
Each row starts with a zero and a one, followed by the coordinates of the vertex. I don’t remember what the zero means. The one means that the subsequent numbers represent a vertex (it is also possible to encode rays and lines in a Vdescription).
Now we get the Hrepresentations with the help of the
scdd
function:
H1 < scdd(V1) H2 < scdd(V2) H3 < scdd(V3) H4 < scdd(V4) H5 < scdd(V5)
Let’s look at a Hdescription:
H1 ## $output ## [,1] [,2] [,3] [,4] [,5] ## [1,] 0 1 2.802517e+00 2.220446e16 1.070466e+00 ## [2,] 0 1 8.881784e16 1.070466e+00 2.802517e+00 ## [3,] 0 1 1.070466e+00 2.802517e+00 9.614813e17 ## [4,] 0 1 1.732051e+00 1.732051e+00 1.732051e+00 ## attr(,"representation") ## [1] "H"
In the first column, a zero indicates that the corresponding rows represents an inequality. The number in the second column is the right hand side of the inequality, and the subsequent three numbers are the coefficients of the linear inequality.
Now, let’s join all the linear inequalities. That is, we stack the five matrices:
H < rbind(H1$output, H2$output, H3$output, H4$output, H5$output)
The scdd
function converts from the Vrepresentation to the
Hrepresentation and viceversa. Hence we get the vertices of the
intersection by applying this function to the above Hrepresentation:
(V < scdd(H) )
## $output ## [,1] [,2] [,3] [,4] [,5] ## [1,] 0 1 2.205282e01 8.212094e16 3.568221e01 ## [2,] 0 1 4.240675e16 3.568221e01 2.205282e01 ## [3,] 0 1 5.654233e16 3.568221e01 2.205282e01 ## [4,] 0 1 2.205282e01 0.000000e+00 3.568221e01 ## [5,] 0 1 3.568221e01 2.205282e01 4.574370e17 ## [6,] 0 1 3.568221e01 2.205282e01 4.574370e17 ## [7,] 0 1 2.205282e01 3.202059e16 3.568221e01 ## [8,] 0 1 1.017762e15 3.568221e01 2.205282e01 ## [9,] 0 1 6.785079e16 3.568221e01 2.205282e01 ## [10,] 0 1 2.205282e01 5.010035e16 3.568221e01 ## [11,] 0 1 3.568221e01 2.205282e01 2.795606e16 ## [12,] 0 1 3.568221e01 2.205282e01 9.718761e17 ## attr(,"representation") ## [1] "V"
The vertices are given in the last three columns. Let’s extract them:
vertices < V$output[, c(3L, 4L, 5L)]
There are twelve vertices. The intersection is a regular icosahedron. Now we will plot it. First, we compute the convex hull of these vertices with the cxhull package. The icosahedron is convex, so its convex hull is itself. But cxhull also provides the faces and the edges that are used for plotting.
library(cxhull) icosahedron < cxhull(vertices, triangulate = TRUE)
There’s a convenient function in cxhull to plot a
convec hull with rgl, namely
plotConvexHull3d
:
library(rgl) open3d(windowRect = c(50, 50, 562, 562)) view3d(10, 80, zoom = 0.7) plotConvexHull3d( icosahedron, palette = hcl.colors(256, "BuPu"), bias = 0.25, edgesColor = "yellow", tubesRadius = 0.06, spheresRadius = 0.08 )
The Python way
Here is the Python code computing the intersection of the five tetrahedra:
import numpy as np import cdd as cdd # the twenty vertices phi = (1 + np.sqrt(5)) / 2 a = 1 / np.sqrt(3) b = a / phi c = a * phi vertices = np.vstack( ( np.array([a, a, a]), np.array([a, a, a]), np.array([a, a, a]), np.array([a, a, a]), np.array([a, a, a]), np.array([a, a, a]), np.array([0, b, c]), np.array([0, b, c]), np.array([0, b, c]), np.array([c, 0, b]), np.array([c, 0, b]), np.array([c, 0, b]), np.array([b, c, 0]), np.array([b, c, 0]), np.array([b, c, 0]), np.array([b, c, 0]), np.array([0, b, c]), np.array([a, a, a]), np.array([c, 0, b]), np.array([a, a, a]), ) ) # tetrahedra vertices tetra1Idxs = [16, 13, 1, 10] tetra2Idxs = [17, 0, 3, 4] tetra3Idxs = [18, 5, 14, 6] tetra4Idxs = [2, 12, 11, 7] tetra5Idxs = [19, 15, 9, 8] th1 = vertices[tetra1Idxs, :] th2 = vertices[tetra2Idxs, :] th3 = vertices[tetra3Idxs, :] th4 = vertices[tetra4Idxs, :] th5 = vertices[tetra5Idxs, :] # make the Vrepresentation of each tetrahedron; you have to prepend the # vertices array with a column of ones v = np.column_stack((np.ones(4), th1)) mat = cdd.Matrix(v, number_type='float') mat.rep_type = cdd.RepType.GENERATOR poly1 = cdd.Polyhedron(mat) v = np.column_stack((np.ones(4), th2)) mat = cdd.Matrix(v, number_type='float') mat.rep_type = cdd.RepType.GENERATOR poly2 = cdd.Polyhedron(mat) v = np.column_stack((np.ones(4), th3)) mat = cdd.Matrix(v, number_type='float') mat.rep_type = cdd.RepType.GENERATOR poly3 = cdd.Polyhedron(mat) v = np.column_stack((np.ones(4), th4)) mat = cdd.Matrix(v, number_type='float') mat.rep_type = cdd.RepType.GENERATOR poly4 = cdd.Polyhedron(mat) v = np.column_stack((np.ones(4), th5)) mat = cdd.Matrix(v, number_type='float') mat.rep_type = cdd.RepType.GENERATOR poly5 = cdd.Polyhedron(mat) # Hrepresentations of the tetrahedra h1 = poly1.get_inequalities() h2 = poly2.get_inequalities() h3 = poly3.get_inequalities() h4 = poly4.get_inequalities() h5 = poly5.get_inequalities() # join the five sets of linear inequalities; this will give the intersection hintersection = np.vstack((h1, h2, h3, h4, h5)) # make the Vrepresentation of the intersection mat = cdd.Matrix(hintersection, number_type='float') mat.rep_type = cdd.RepType.INEQUALITY polyintersection = cdd.Polyhedron(mat) # get the vertices; they are given in a matrix prepended by a column of ones vintersection = polyintersection.get_generators() # get rid of the column of ones n_inequalities = vintersection.row_size intersection = np.array([ vintersection[i][1:4] for i in range(n_inequalities) ])
Results are the same as the ones we got with R:
array([[2.20528179e01, 8.21209393e16, 3.56822090e01], [ 4.24067460e16, 3.56822090e01, 2.20528179e01], [ 5.65423280e16, 3.56822090e01, 2.20528179e01], [ 2.20528179e01, 0.00000000e+00, 3.56822090e01], [ 3.56822090e01, 2.20528179e01, 4.57437043e17], [ 3.56822090e01, 2.20528179e01, 4.57437043e17], [ 2.20528179e01, 3.20205930e16, 3.56822090e01], [ 1.01776190e15, 3.56822090e01, 2.20528179e01], [ 6.78507937e16, 3.56822090e01, 2.20528179e01], [2.20528179e01, 5.01003463e16, 3.56822090e01], [3.56822090e01, 2.20528179e01, 2.79560644e16], [3.56822090e01, 2.20528179e01, 9.71876138e17]])
It is possible to get the edges of the polyhedron with
pycddlib; see how I did in
this post. And to get the convex hull with its faces, one can use
scipy.spatial.ConvexHull(intersection)
.
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