---
title: "Uniform sampling in a convex hull"
author: "Stéphane Laurent"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Uniform sampling in a convex hull}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

As an illustration of the `uniformly` package, we will show how to 
uniformly sample some points in a convex hull.

We give an illustration in dimension 3 (in dimension 2, use the function 
`runif_in_polygon`). 

Let's store the vertices of an icosahedron in a matrix `vs`:

```{r}
vs <- t(rgl::icosahedron3d()$vb[1:3,])
head(vs)
```

The icosahedron is convex, therefore its convex hull is itself. 

The `delaunayn` function of the `geometry` package calculates a "triangulation" 
(*tetrahedralization* in dimension 3) of the convex hull of a set of points. 
We use it to get a tetrahedralization of our icoshaedron:

```{r, message=FALSE}
library(geometry)
tetrahedra <- delaunayn(vs, options="Qz")
head(tetrahedra)
```

Each row of the `tetrahedra` matrix is a vector of four identifiers of the 
vertices defining a tetrahedron.

Now, we calculate the volumes of each of these tetrahedra with the 
`volume_tetrahedron` function:

```{r}
library(uniformly)
volumes <- 
  apply(tetrahedra, 1, 
        function(t){
          volume_tetrahedron(vs[t[1],], vs[t[2],], vs[t[3],], vs[t[4],])
        })
```

We normalize the volumes:

```{r}
probs <- volumes/sum(volumes)
```

Now, here is the algorithm to uniformly sample a point in the icosahedron:

- select a tetrahedron at random, with probability given by the normalized volumes;

- uniformly sample a point in the picked tetrahedron.

That is:

```{r}
i <- sample.int(nrow(tetrahedra), 1, prob=probs)
th <- tetrahedra[i,]
runif_in_tetrahedron(1, vs[th[1],], vs[th[2],], vs[th[3],], vs[th[4],])
```

Let's use the algorithm to sample 100 random points:

```{r}
nsims <- 100
sims <- matrix(NA_real_, nrow=nsims, ncol=3)
for(k in 1:nsims){
  th <- tetrahedra[sample.int(nrow(tetrahedra), 1, prob=probs),]
  sims[k,] <- runif_in_tetrahedron(1, vs[th[1],], vs[th[2],], vs[th[3],], vs[th[4],])
}
```

```{r}
library(rgl)
open3d(windowRect=c(100,100,600,600))
shade3d(icosahedron3d(), color="red", alpha=0.3)
points3d(sims)
rglwidget()
```

We can proceed in the same way in higher dimension, using the functions 
`volume_simplex` and `runif_in_simplex` instead of the functions 
`volume_tetrahedron` and `runif_in_tetrahedron`.


## Sampling from a triangulated object

Note that the convexity is not the *sine qua non* condition to apply the 
above procedure: the ingredient we need is the "triangulation" of the object. 
We took a convex shape because `delaunayn` provides the triangulation of a 
convex shape. 

Let's give an example for a 3D star. Here is the star:

```{r}
vs <- rbind(
  c(7.889562, 1.150329, -2.173651),
  c(2.212808, 1.150329, -2.230414),
  c(0.068023, 1.150328, -7.923502),
  c(-2.151306, 1.150329, -2.254857),
  c(-7.817406, 1.150328, -2.261558),
  c(-3.523133, 1.150328, 1.888122),
  c(-4.869315, 1.150328, 6.987552),
  c(-0.006854, 1.150329, 4.473047),
  c(4.838127, 1.150328, 7.041885),
  c(3.538153, 1.150329, 1.927652),
  c(0.033757, 0.000000, -0.314657),
  c(0.035668, 2.269531, -0.312831)
)
faces <- rbind(
  c(1, 11, 2),
  c(2, 11, 3),
  c(3, 11, 4),
  c(4, 11, 5),
  c(5, 11, 6),
  c(6, 11, 7),
  c(7, 11, 8),
  c(8, 11, 9),
  c(9, 11, 10),
  c(10, 11, 1),
  c(1, 12, 10),
  c(10, 12, 9),
  c(9, 12, 8),
  c(8, 12, 7),
  c(7, 12, 6),
  c(6, 12, 5),
  c(5, 12, 4),
  c(4, 12, 3),
  c(3, 12, 2),
  c(2, 12, 1)
)
open3d(windowRect=c(100,100,600,600))
for(i in 1:nrow(faces)){
 triangles3d(rbind(
   vs[faces[i,1],], 
   vs[faces[i,2],], 
   vs[faces[i,3],]), 
   color="red", alpha=0.4)
}
rglwidget()
```

This star is not convex but it is star-shaped with respect to its centroid, 
and its faces are triangular. 
Therefore we get a tetrahedralization by joining the centroid to each of the 
triangular faces. 

Let's calculate the volumes of these tetrahedra:

```{r}
centroid <- colMeans(vs)
volumes <- apply(faces, 1,function(f){
  volume_tetrahedron(vs[f[1],], vs[f[2],], vs[f[3],], centroid)
})
probs <- volumes/sum(volumes)
```

Now we pick a face at random, with probability given by the normalized volumes, 
and we sample in the corresponding tetrahedron:

```{r}
nsims <- 500
sims <- matrix(NA_real_, nrow=nsims, ncol=3)
for(k in 1:nsims){
  f <- faces[sample.int(nrow(faces), 1, prob=probs),]
  sims[k,] <- runif_in_tetrahedron(1, vs[f[1],], vs[f[2],], vs[f[3],], centroid)
}
```

And now, let's add the sampled points:

```{r}
points3d(sims)
rglwidget()
```

