# Clipping an isosurface to a ball, and more

**Saturn Elephant**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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We will firstly show how to clip an isosurface to a ball with R, and then, more generally, how to clip a surface to an arbitrary region. In the last part we show how to achieve the same with Python.

# Using R

## The Togliatti isosurface, clipped to a box

The Togliatti surface is the isosurface \(f(x, y, z) = 0\), where the function \(f\) is defined as follows in R:

f <- function(x,y,z){ 64*(x-1)* (x^4-4*x^3-10*x^2*y^2-4*x^2+16*x-20*x*y^2+5*y^4+16-20*y^2) - 5*sqrt(5-sqrt(5))*(2*z-sqrt(5-sqrt(5)))*(4*(x^2+y^2-z^2)+(1+3*sqrt(5)))^2 }

To plot an isosurface in R, there is the
**misc3d** package. Below we plot the Togliatti surface
clipped to the box
\([-5,5] \times [-5,5] \times [-4,4]\).

library(misc3d) # make grid nx <- 220L; ny <- 220L; nz <- 220L x <- seq(-5, 5, length.out = nx) y <- seq(-5, 5, length.out = ny) z <- seq(-4, 4, length.out = nz) G <- expand.grid(x = x, y = y, z = z) # calculate voxel voxel <- array(with(G, f(x, y, z)), dim = c(nx, ny, nz)) # compute isosurface surf1 <- computeContour3d(voxel, maxvol = max(voxel), level = 0, x = x, y = y, z = z) # make triangles triangles <- makeTriangles(surf1, smooth = TRUE)

drawScene.rgl(triangles, color = "yellowgreen")

It is fun, but you will see later that it is prettier when clipped to a ball. And it is desirable to get rid of the isolated components at the top.

The simplest way to clip the surface to a ball consists in using the
`mask`

argument of the function
`computeContour3d`

. We use
\(4.8\) as the radius.

mask <- array(with(G, x^2+y^2+z^2 < 4.8^2), dim = c(nx, ny, nz)) surf2 <- computeContour3d( voxel, maxvol = max(voxel), level = 0, mask = mask, x = x, y = y, z = z ) triangles <- makeTriangles(surf2, smooth = TRUE) drawScene.rgl(triangles, color = "orange")

That’s not bad, but you see the problem: the borders are not smooth.

## Resorting to spherical coordinates

Using spherical coordinates will allow us to get the surface clipped to the ball with smooth borders. Here is the method:

# Togliatti surface equation with spherical coordinates h <- function(ρ, θ, ϕ){ x <- ρ * cos(θ) * sin(ϕ) y <- ρ * sin(θ) * sin(ϕ) z <- ρ * cos(ϕ) f(x, y, z) } # make grid nρ <- 300L; nθ <- 300L; nϕ <- 300L ρ <- seq(0, 4.8, length.out = nρ) # ρ runs from 0 to the desired radius θ <- seq(0, 2*pi, length.out = nθ) ϕ <- seq(0, pi, length.out = nϕ) G <- expand.grid(ρ=ρ, θ=θ, ϕ=ϕ) # calculate voxel voxel <- array(with(G, h(ρ, θ, ϕ)), dim = c(nρ, nθ, nϕ)) # calculate isosurface surf3 <- computeContour3d(voxel, maxvol = max(voxel), level = 0, x = ρ, y = θ, z = ϕ) # transform to Cartesian coordinates surf3 <- t(apply(surf3, 1L, function(ρθϕ){ ρ <- ρθϕ[1L]; θ <- ρθϕ[2L]; ϕ <- ρθϕ[3L] c( ρ * cos(θ) * sin(ϕ), ρ * sin(θ) * sin(ϕ), ρ * cos(ϕ) ) })) # draw isosurface drawScene.rgl(makeTriangles(surf3, smooth=TRUE), color = "violetred")

Now the surface is pretty nice. The borders are smooth.

## Another way: using the rgl package

Nowadays, in the **rgl** package, there is the function
`clipMesh3d`

which allows to
*clip a mesh to a general region*. In order to use it, we need a
`mesh3d`

object. There is an unexported function in
**misc3d** which allows to easily get a
`mesh3d`

object; it is called `t2ve`

.

We start with the isosurface clipped to the box:

triangles <- makeTriangles(surf1) # convert to rgl `mesh3d` mesh <- misc3d:::t2ve(triangles) rglmesh <- tmesh3d(mesh$vb, mesh$ib)

To define the region to which we want to clip the mesh, one has to introduce a function and to give a bound for this function. Here the region is \(x^2 + y^2 + z^2 < 4.8^2\), so we proceed as follows:

fn <- function(x, y, z) x^2 + y^2 + z^2 # or function(xyz) rowSums(xyz^2) clippedmesh <- addNormals(clipMesh3d( rglmesh, fn, bound = 4.8^2, greater = FALSE ))

It is a bit slow. Probably the algorithm is not very easy. But we get our desired result:

shade3d(clippedmesh, color = "magenta")

# Using Python

With the wonderful Python library **PyVista**, we proceed
as follows to create the mesh of the Togliatti isosurface:

from math import sqrt import numpy as np import pyvista as pv def f(x, y, z): return ( 64 * (x - 1) * ( x ** 4 - 4 * x ** 3 - 10 * x ** 2 * y ** 2 - 4 * x ** 2 + 16 * x - 20 * x * y ** 2 + 5 * y ** 4 + 16 - 20 * y ** 2 ) - 5 * sqrt(5 - sqrt(5)) * (2 * z - sqrt(5 - sqrt(5))) * (4 * (x ** 2 + y ** 2 - z ** 2) + (1 + 3 * sqrt(5))) ** 2 ) # generate data grid for computing the values X, Y, Z = np.mgrid[(-5):5:250j, (-5):5:250j, (-4):4:250j] # create a structured grid grid = pv.StructuredGrid(X, Y, Z) # compute and assign the values values = f(X, Y, Z) grid.point_data["values"] = values.ravel(order="F") # compute the isosurface f(x, y, z) = 0 isosurf = grid.contour(isosurfaces=[0]) mesh = isosurf.extract_geometry()

To plot the mesh clipped to the box:

# surface clipped to the box: mesh.plot(smooth_shading=True, color="yellowgreen", specular=15)

The `remove_points`

method is similar to the
`mask`

approach with **misc3d** (it produces
non-smooth borders):

# surface clipped to the ball of radius 4.8, with the help of `remove_points`: lengths = np.linalg.norm(mesh.points, axis=1) toremove = lengths >= 4.8 masked_mesh, idx = mesh.remove_points(toremove) masked_mesh.plot(smooth_shading=True, color="orange", specular=15)

To get smooth borders, you have to use the
`clip_scalar`

method:

# surface clipped to the ball of radius 4.8, with the help of `clip_scalar`: mesh["dist"] = lengths clipped_mesh = mesh.clip_scalar("dist", value=4.8) clipped_mesh.plot( smooth_shading=True, cmap="inferno", window_size=[512, 512], show_scalar_bar=False, specular=15, show_axes=False, zoom=1.2, background="#363940" )

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