# Enneper surface with rotating checkerboard

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

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The **github** branch of my Github repository
cgalMeshes has a
vignette explaining how to use *parameterizations* of surface
meshes. A parameterization allows to map a texture on a mesh. Some of
them are *conformal*, meaning that they preserve the angles (the
best they can). To install this branch with the vignette, run

remotes::install_github( "stla/cgalMeshes@github", dependencies = TRUE, build_vignettes = TRUE )

This may take a while. And if that doesn’t work and you use Windows, you can try to install the binary zip file located at https://github.com/stla/cgalMeshes/releases/tag/Github_v3.1.0.

The vignette shows a couple of examples of parametrizations. Among them, there is the Enneper surface with a radial checkberboard:

Here I will show how one can rotate the radial checkerboard.

Let’s start by the beginning. We make the Enneper mesh.

library(cgalMeshes) library(rgl) # Enneper surface parameteritazion #### n <- 3 Enneper <- function(phi, r) { rbind( r*cos(phi) - r^(2*n-1)*cos((2*n-1)*phi)/(2*n-1), r*sin(phi) + r^(2*n-1)*sin((2*n-1)*phi)/(2*n-1), 2*r^n*cos(n*phi)/n ) } # do the mesh #### rmesh <- parametricMesh( Enneper, urange = c(0, 2*pi), vrange = c(0, 1.3), periodic = c(TRUE, FALSE), nu = 512, nv = 512, clean = TRUE )

Now we convert it to a CGAL mesh:

# convert to CGAL mesh #### mesh <- cgalMesh$new(rmesh)

There’s not enough vertices in this mesh; if we use this one, the checkerboard will not have regular lines. So we perform an isotropic remeshing to add some vertices:

# add vertices in order that the checkerboard has regular lines #### mesh$isotropicRemeshing(0.01, iterations = 3, relaxSteps = 2)

Now we compute the mesh parameterization:

# compute mesh parameterization #### UV <- mesh$parameterization(method = "DCP", UVborder = "circle")

And here is the code to do the radial checkerboard:

# radial checkerboard #### UV0 <- UV UV <- 10 * (UV0 - 0.5) radii <- sqrt(apply(UV, 1L, crossprod)) angles <- 10 * (1 + atan2(UV[, 2L], UV[, 1L])/pi) clrs <- ifelse( floor(radii) %% 2 == 0, ifelse( floor(angles) %% 2 == 0, "navy", "yellow" ), ifelse( floor(angles) %% 2 == 0, "yellow", "navy" ) ) # check the checkerboard is correct #### plot( UV0, type = "p", asp = 1, pch = ".", col = clrs, xlab = "u", ylab = "v", xlim = c(0,1), ylim = c(0,1) )

We compute the normals, we convert the mesh to a
**rgl** mesh, and we assign the checkerboard colors:

# compute normals, convert to 'rgl' mesh, and add colors #### mesh$computeNormals() rmesh <- mesh$getMesh() rmesh$material <- list(color = clrs)

Now we make the animation, by rotating the checkerboard:

# animation rotating checkboard #### fclrs <- function(alpha) { tests <- floor(angles + alpha) %% 2 == 0 ifelse( floor(radii) %% 2 == 0, ifelse( tests, "navy", "yellow" ), ifelse( tests, "yellow", "navy" ) ) } # make animation frames #### alpha_ <- seq(0, 2, length.out = 19L)[-1L] open3d(windowRect = 50 + c(0, 0, 512, 512)) view3d(0, -20, zoom = 0.7) for(i in seq_along(alpha_)) { clrs <- fclrs(alpha_[i]) rmesh$material <- list(color = clrs) shade3d(rmesh, meshColor = "vertices") snapshot3d(sprintf("zzpic%03d.png", i), webshot = FALSE) clear3d() }

It remains to mount the animation with **gifski**:

# mount animation #### library(gifski) gifski( png_files = Sys.glob("zzpic*.png"), gif_file = "Enneper-radialCheckerboard-rotating.gif", width = 512, height = 512, delay = 1/8 )

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**Saturn Elephant**.

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