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 \(\pi\)).

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:

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).

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:

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:

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 functions are a bit slow. My package RCGAL (not on CRAN) can construct an elevated Delaunay tessellation, and it is faster.

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

You can directly get the volume:

del[["volume"]]
## [1] 3.141593

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

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)

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):

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
) 

library(rgl)
persp3d(del, front = "lines", back = "lines", col = "blue")
aspect3d(2, 2, 1)