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## 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)

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