---
title: "Hyperbolic gircope - using 'cxhull' and 'gyro'"
author: "Stéphane Laurent"
date: '2022-02-28'
tags: R, maths, geometry, rgl, graphics
rbloggers: yes
output:
md_document:
variant: markdown
preserve_yaml: true
html_document:
highlight: kate
keep_md: no
highlighter: pandoc-solarized
---
```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, collapse = TRUE)
```
This post is a demonstration of the **cxhull** and **gyro** packages.
I will use these packages (and others) to draw an hyperbolic version
of the stereographic projection of a convex 4D polytope, the *gircope* or
*great rhombicuboctahedral prism*. The gircope has twenty-eight cells, but
I will only draw the twelve cubes among them.
As said in
[this wiki](https://polytope.miraheze.org/wiki/Great_rhombicuboctahedral_prism),
the vertices of the gircope are given by all permutations of the first three
coordinates of:
$$
\left(\pm\frac{1+2\sqrt{2}}{2}, \pm\frac{1+\sqrt{2}}{2},
\pm\frac{1}{2}, \pm\frac{1}{2}\right).
$$
I define these vertices in R as follows:
```{r vertices4D}
library(gyro) # to use the `changesOfSign` function
library(arrangements) # to use the `permutations` function
x <- c(
(1 + 2*sqrt(2)) / 2,
(1 + sqrt(2)) / 2,
1/2
)
vertices <- changesOfSign(
cbind(
t(apply(permutations(3L), 1L, function(perm) x[perm])),
1/2
)
)
```
Obviously, the vertices of the gircope lie on a sphere centered at the origin:
```{r crossprod}
apply(vertices, 1L, crossprod)
```
We will need the value of the radius of this sphere later, for the
stereographic projection:
```{r radius}
R <- sqrt(c(crossprod(vertices[1L, ])))
```
The gircope is convex, hence it equals its convex hull. But we have only its
vertices so far, and we need its edges, its cells, and its ridges (faces of
the cells). This is why we use **cxhull** now:
```{r cxhull}
library(cxhull)
hull <- cxhull(vertices)
edges <- hull[["edges"]]
cells <- hull[["facets"]]
ridges <- hull[["ridges"]]
```
A cube has eight vertices, and among the cells of the gircope, only the cubic
ones have eight vertices. So we get all the cubic cells like this:
```{r cubicCells}
cubicCells <-
Filter(function(cell) length(cell[["vertices"]]) == 8L, cells)
```
No we need the faces of the cubes (squares). We can easily get the indices of
their vertices but we have to order them. That's what the `polygonize` function
below does:
```{r polygonize}
polygonize <- function(edges){
nedges <- nrow(edges)
indices <- edges[1L, ]
i <- indices[2L]
edges <- edges[-1L, ]
for(. in 1L:(nedges-2L)){
j <- which(apply(edges, 1L, function(e) i %in% e))
i <- edges[j, ][which(edges[j, ] != i)]
indices <- c(indices, i)
edges <- edges[-j, ]
}
indices
}
```
Now we can get the indices of the vertices of the squares:
```{r squares}
squares <- t(vapply(
do.call(c, lapply(cubicCells, `[[`, "ridges")),
function(r) polygonize(ridges[[r]][["edges"]]),
integer(4L)
))
```
Now, let's project the 4D vertices to the 3D space, with a stereographic
projection:
```{r stereoproj}
verts3D <- t(apply(vertices, 1L, function(v){
v[1L:3L] / (R - v[4L])
}))
```
We are ready for plotting. We can't directly draw hyperbolic squares with the
**gyro** package. It only allows to draw hyperbolic triangles, with the
`gyrotriangle` function. So we draw an hyperbolic square by splitting it into
two triangles, we merge these two triangles with `Morpho::mergeMeshes` and
we remove the duplicated vertices of the resulting mesh with `Rvcg::vcgClean`.
```{r rgl, eval=FALSE}
library(rgl)
library(Morpho) # to use the `mergeMeshes` function
library(Rvcg) # to use the `vcgClean` function
s <- 0.5 # hyperbolic curvature
open3d(windowRect = c(50, 50, 562, 562), zoom = 0.8)
bg3d(rgb(54, 57, 64, maxColorValue = 255))
for(i in 1L:nrow(squares)){
square <- squares[i, ]
mesh1 <- gyrotriangle(
verts3D[square[1L], ], verts3D[square[2L], ], verts3D[square[3L], ],
s = s
)
mesh2 <- gyrotriangle(
verts3D[square[1L], ], verts3D[square[3L], ], verts3D[square[4L], ],
s = s
)
mesh <- vcgClean(mergeMeshes(mesh1, mesh2), sel = c(0, 7), silent = TRUE)
shade3d(mesh, color = "violetred")
}
for(i in 1L:nrow(edges)){
edge <- edges[i, ]
A <- verts3D[edge[1L], ]; B <- verts3D[edge[2L], ]
tube <- gyrotube(A, B, s = s, radius = 0.025)
shade3d(tube, color = "whitesmoke")
}
spheres3d(verts3D, radius = 0.03, color = "whitesmoke")
```
![](figures/hyperbolicGircope.gif)
To make the animation, I used the following code.
```{r anim, eval=FALSE}
movie3d(
spin3d(axis = c(1, 1, 0), rpm = 10),
duration = 6, fps = 10,
movie = "pic", dir = ".",
convert = FALSE,
startTime = 1/10,
webshot = FALSE)
```
This code produces the files **pic001.png**, ... **pic060.png**. Then I
assembled them into a GIF with
[gifski](https://laustep.github.io/stlahblog/posts/gifskiBash.html) (you can
use **ImageMagick** instead).