---
author: Stéphane Laurent
date: '2022-02-26'
highlighter: 'pandoc-solarized'
output:
  html_document:
    highlight: kate
    keep_md: no
  md_document:
    preserve_yaml: True
    variant: markdown
rbloggers: yes
tags: 'R, maths, geometry, rgl'
title: Volume under surface from points
---

The problem
-----------

Suppose you want to get the volume under a surface but you only have
some points on this surface. For the illustration, I will take the
surface defined by $z = \exp\bigl(-(x^2 + y^2)\bigr)$ on the square
$[-5, 5] \times [-5, 5]$. Then the volume we're looking for is close to
$\pi$ (the integral on $[-\infty, +\infty] \times [-\infty, +\infty]$ is
exactly $\pi$).

``` {.r}
f <- function(x, y){
  exp(-(x*x + y*y))
}
```

Now let's define a grid on $[-5, 5] \times [-5, 5]$ and the value of $z$
for each point on this grid:

``` {.r}
x <- seq(-5, 5, length.out = 100)
y <- seq(-5, 5, length.out = 100)
grd <- transform( # data (x_i, y_i, z_i)
  expand.grid(x = x, y = y), z = f(x, y)
) 
```

Elevated Delaunay tessellation - using 'deldir'
-----------------------------------------------

A solution consists in constructing a Delaunay tessellation of the
surface and then to sum the volumes under the Delaunay triangles. The
**deldir** package allows to construct such a Delaunay tessellation
(which I call an *elevated Delaunay tessellation*).

``` {.r}
library(deldir)
del <- deldir( # Delaunay
  x = grd[["x"]], y = grd[["y"]], z = grd[["z"]],
  rw = c(-5, 5, -5, 5), round = FALSE
) 
trgls <- triang.list(del) # extracts all triangles
```

The function below calculates the volume under a triangle:

``` {.r}
volume_under_triangle <- function(trgl){
  with(
    trgl, 
    sum(z) * 
      (x[1L]*y[2L] - x[2L]*y[1L] + x[2L]*y[3L] -
         x[3L]*y[2L] + x[3L]*y[1L] - x[1L]*y[3L]) / 6
  )
}
```

So here is our approximation of the volume:

``` {.r}
volumes <- vapply(trgls, volume_under_triangle, numeric(1L))
sum(volumes)
## [1] 3.141592
```

Using 'RCGAL'
-------------

If you ran the above code, you noticed that the `deldir` function as
well as the `triang.list` function are a bit slow. My package
[RCGAL](https://laustep.github.io/stlahblog/posts/SurfaceReconstruction.html)
(not on CRAN) can construct an elevated Delaunay tessellation, and it is
faster.

``` {.r}
library(RCGAL)
points <- as.matrix(grd)
del <- delaunay(points, elevation = TRUE)
```

You can directly get the volume:

``` {.r}
del[["volume"]]
## [1] 3.141593
```

And you can easily plot the elevated Delaunay tessellation with the help
of the **rgl** package:

``` {.r}
mesh <- del[["mesh"]]
library(rgl)
open3d(windowRect = c(50, 50, 562, 306), zoom = 0.5)
aspect3d(1, 1, 3)
shade3d(mesh, color = "limegreen")
wire3d(mesh)
```

![](figures/rgl_elevated_delaunay.png)

Update 2022-03-02: using 'tessellation'
---------------------------------------

The elevated Delaunay tessellation is now available in my package
[tessellation](https://github.com/stla/tessellation). The command to get
it is the same as the 'RCGAL' command and the output is similar.

``` {.r}
del <- tessellation::delaunay(points, elevation = TRUE)
del[["volume"]]
## [1] 3.141593
```

Update 2022-03-10: using 'RCDT'
-------------------------------

The elevated Delaunay triangulation is now available in my package
[RCDT](https://github.com/stla/RCDT).

``` {.r}
del <- RCDT::delaunay(points, elevation = TRUE)
del[["volume"]]
## [1] 3.141593
```

Interactive plot with 'deldir'
------------------------------

The **deldir** also allows to get an interactive graphic from the
elevated Delaunay tessellation. This requires the **rgl** package. I do
it below with a less fine grid, otherwise the visualization is not nice
(too dense):

``` {.r}
x <- seq(-3, 3, length.out = 20)
y <- seq(-3, 3, length.out = 20)
grd <- transform(
  expand.grid(x = x, y = y), z = f(x, y)
) 
del <- deldir(
  x = grd[["x"]], y = grd[["y"]], z = grd[["z"]],
  rw = c(-3, 3, -3, 3), round = FALSE
) 
```

``` {.r}
library(rgl)
persp3d(del, front = "lines", back = "lines", col = "blue")
aspect3d(2, 2, 1)
```

![](figures/deldir_elevated_delaunay.gif)