[This article was first published on Saturn Elephant, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't. In this post, I provide some R code which generates a mesh of the Mandelbulb, a well-known 3D fractal.
The Mandelbulb is an isosurface, and I use the rmarchingcubes package to get a mesh of this isosurface. Since the Mandelbulb has many details, a thin grid of the voxel space is necessary, and that is why I use Rcpp to generate the voxel. Here is the C++ code:
// file mandelbulb.cpp#include <Rcpp.h>using namespace Rcpp;double mandelbulb0( double x, double y, double z, const unsigned power, const double phase) { const double x0 = x; const double y0 = y; const double z0 = z; const double k = power; double r, rkm1, rk, theta, phi; double dr = 1.0; int i; for(i = 0; i < 10; i++) { r = sqrt(x*x + y*y + z*z); if(r > 2) { return 2.0 * r * log(r) / dr; } rkm1 = pow(r, k - 1.0); dr = k * rkm1 * dr + 1.0; theta = k * atan2(sqrt(x*x + y*y), z) + phase; phi = k * atan2(y, x); rk = rkm1 * r; x = rk * cos(phi) * sin(theta) + x0; y = rk * sin(phi) * sin(theta) + y0; z = rk * cos(theta) + z0; } return 0.0;}// [[Rcpp::export]]NumericVector mandelbulb( const double m, const double M, const unsigned n, const unsigned power, const double phase) { NumericVector out(n * n * n); const double h = (M - m) / (n - 1); double x, y, z; unsigned i, j, k; unsigned l = 0; for(i = 0; i < n; i++) { x = m + i*h; for(j = 0; j < n; j++) { y = m + j*h; for(k = 0; k < n; k++) { z = m + k*h; out(l) = mandelbulb0(x, y, z, power, phase); l++; } } } out.attr("dim") = Dimension(n, n, n); return out;}
In fact, there are several Mandelbulb, each corresponding to a value of the power argument in the above code. The most popular one is the one with power=8. At the end of this post, I’ll show you the effect of the phase argument.
Now here is the R code which generates the mesh:
Rcpp::sourceCpp("mandelbulb.cpp")library(rmarchingcubes)library(rgl)n <- 512L # more than enoughx <- y <- z <- seq(-1.2, 1.2, length.out = n)voxel <- mandelbulb(-1.2, 1.2, n, 8L, 0)ctr <- contour3d(voxel, level = 0.01, x = x, y = y, z = z)mesh <- tmesh3d( vertices = t(ctr[["vertices"]]), indices = t(ctr[["triangles"]]), normals = ctr[["normals"]], homogeneous = FALSE)
This mesh can be plotted with rgl. But let’s add some color before. I like the ‘klingon’ color palette of the trekcolors package.
library(trekcolors)fpalette <- colorRamp(trek\_pal("klingon", reverse = TRUE))d2 <- apply(mesh[["vb"]][-4L, ], 2L, crossprod)d2 <- (d2 - min(d2)) / diff(range(d2))RGB <- fpalette(d2)mesh[["material"]] <- list( "color" = rgb(RGB[, 1L], RGB[, 2L], RGB[, 3L], maxColorValue = 255))open3d(windowRect = c(50, 50, 562, 562), zoom = 0.7)shade3d(mesh, shininess = 128)
The animation below shows the effect of the phase argument, varying from (0) to (2\pi):
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