---
title: "Update of the 'cgalMeshes' package"
author: "Stéphane Laurent"
date: '2023-01-28'
tags: R, geometry, C++, Rcpp, rgl
rbloggers: yes
output:
  md_document:
    variant: markdown
    preserve_yaml: true
  html_document:
    highlight: kate
    keep_md: no
highlighter: pandoc-solarized
---

I made some progress on the **cgalMeshes** package, that I introduced 
[here](https://laustep.github.io/stlahblog/posts/cgalMeshes.html). The new 
version is not on CRAN yet.

In particular, one can know assign some properties to the meshes (the 
`cgalMesh` objects). One can assign some face properties: colors and scalars, 
and some vertex properties: colors, scalars, and normals. Actually a color can 
be any character string, but if you really set some colors, and you write the 
mesh to a file, then you'll retrieve the colors in this file. Some of these 
properties are preserved by certain operations on the mesh. For example, all 
properties are preserved when one splits a mesh into connected components. 
This was my original motivation by the way, more precisely I wanted to retrieve 
the normals of the mesh in its connected components, because I had a mesh made 
with the **rmarchingcubes** package and then it had very good normals (the 
normals returned by the **rmarchingcubes** package are obtained by an 
approximation of the gradient of the function defining the isosurface). The 
union of two meshes preserves the face properties of the two meshes. This is 
nice when one does not want to lose the colors.

Here is a couple of new features. Let's start by the new methods of the 
`cgalMesh` objects. The `$clip()` method now returns two meshes: the clipped 
mesh and the modified clipping mesh (it is modified by a corefinement). These 
two meshes inherit the colors of the original meshes. For example, here is the 
result one obtains by clipping a cylinder to another cylinder, and then 
clipping the resulting mesh by a third cylinder, with different colors for the 
three cylinders:

![](figures/cgalMeshes_clipCylinders.gif){width=50%}

See [this page by Paul Bourke](http://paulbourke.net/geometry/cylinders/) for 
more fun with intersecting cylinders. I've tried to do something similar to 
the intersection of a bunch of random cylinders but my implementation is too 
time-consuming.

The `$fillBoundaryHole()` method allows to fill the holes of a mesh. There's 
one option, `fair`, allowing to request a smooth hole filling or a flat hole 
filling. Here is an example of a sphere with a hole, filled with the option 
`fair=TRUE` and `fair=FALSE`:

![](figures/cgalMeshes_holefilling.png){width=75%}

Also some subdivision and deformation methods are available now. Below is a 
coarse mesh of a Hopf torus and the mesh obtained after two iterations of the 
Loop subdivision:

![](figures/cgalMeshes_Hopf_LoopSubdiv.png){width=65%}

Now let's see some new functions of the package. The `algebraicMesh()` function 
allows to construct a mesh of an algebraic surface, that is to say an 
isosurface of the form $P(x, y, z) = 0$ where $P$ is a polynomial. Of course 
it is possible to make an isosurface $f(x, y, z) = 0$ where $f$ is arbitrary 
with **CGAL** but I don't know an efficient way to pass a function from R to 
C++. Here is the Barth decic obtained with this function:

![](figures/cgalMeshes_BarthDecic.gif){width=50%}

The algorithm used by **CGAL** is not the marching cubes algorithm. Actually, 
so far, I didn't observe any advantage of this algorithm as compared to the 
marching cubes algorithm (if you know such an advantage, please leave an 
[answer here](https://scicomp.stackexchange.com/q/42410/14840)).

There is now the `convexHull()` function in the package, and also 
`convexHullsIntersection` to (fastly) compute a mesh of the intersection of 
some convex hulls. I used it to get the intersection of the 
[compound of five tetrahedra](https://laustep.github.io/stlahblog/posts/tetrahedraCompound.html):

![](figures/cgalMeshes_cxhullinter.png){width=50%}

The `sphericalTriangle()` function allows to get a mesh of a spherical 
triangle. Here is a spherical icosahedron:

![](figures/cgalMeshes_sphericalIcosahedron.gif){width=50%}

Hopefully the package will pass the CRAN checks.