started blog post Enneper-checkerboard2

inst/essais/Enneper-checkerboard2/Enneper-checkerboard2.Rmd

@@ -15,3 +15,135 @@ I showed how to use the **CGAL** parameterizations ported to my package
 **cgalMeshes** to decorate the Enneper surface of order three with a radial 
 checkerboard.
 
+Here I will show how to map an ordinary, square checkeboard on the Enneper 
+surface of order two.
+
+```{r EnneperMesh, message=FALSE}
+library(cgalMeshes)
+library(rgl)
+
+# Enneper parameterization
+n <- 2
+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
+  )
+}
+# make mesh
+rmesh <- parametricMesh(
+  Enneper, urange = c(0, 2*pi), vrange = c(0, 1.2),
+  periodic = c(TRUE, FALSE), nu = 512L, nv = 128L, clean = TRUE
+)
+# we extract the boundary; we will plot it multiple times
+bndry <- getBoundary3d(rmesh, sorted = TRUE, color = "black")
+```
+
+```{r plotEnneper, eval=FALSE}
+# plot the mesh
+open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.8)
+shade3d(rmesh, col = "deeppink4")
+shade3d(bndry, lwd = 4)
+```
+
+![](./figures/EnneperOrderTwo.png)
+
+Now we add some vertices to the mesh in order that the lines of the 
+checkerboard that we will make later will be straight enough. We use the 
+isotropic remeshing to do so, which does not change the shape of the mesh.
+```{r isotropicRemeshing}
+# convert to CGAL mesh ####
+mesh <- cgalMesh$new(rmesh)
+# take a look at the edge lengths
+edges <- mesh$getEdges()
+summary(edges[["length"]])
+# add vertices in order that the checkeboard will have smooth lines ####
+mesh$isotropicRemeshing(0.008, iterations = 3, relaxSteps = 2)
+```
+
+We first compute a DCP parameterization to make a checkerboard.
+
+```{r DCPnoCorners}
+# compute mesh DCP parameterization ####
+UV <- mesh$parameterization(method = "DCP")
+# the UV-space is the square [0,1]x[0,1]
+# make square checkerboard with 5 squares x 5 squares ####
+checkerboard <- ifelse(
+  (floor(5 * UV[, 1L]) %% 2) == (floor(5 * UV[, 2L]) %% 2), 
+  "gold", "magenta4"
+)
+# add normals, convert to 'rgl' mesh, and add colors ####
+mesh$computeNormals()
+rmesh <- mesh$getMesh()
+rmesh[["material"]] <- list("color" = checkerboard)
+```
+
+```{r plotDCPcheckerboardNoCorners}
+# plot ####
+open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.8)
+bg3d("#363940")
+shade3d(rmesh, meshColor = "vertices", polygon_offset = 1)
+shade3d(b, lwd = 4)
+```
+
+![](./figures/EnneperCheckerboard_NoCorners.png)
+
+This mapped checkerboard is not very nice: the squares are not "aligned" 
+with the mesh. The goal of the rest of this article is to construct a 
+mapped checkerboard "aligned" with the mesh.
+
+
+### First method: DCP with four fixed corners
+
+The DCP parameterization allows to choose four vertices of the mesh which 
+will be the four corners of the square space of the parameterization 
+(the "UV-space"). 
+We choose the four vertices which attain the extreme values of the $x$ 
+coordinate:
+
+```{r theFourCorners, eval=FALSE}
+vs <- mesh$getVertices()
+vsx <- vs[, 1L]
+oinc <- order(vsx)
+odec <- order(vsx, decreasing = TRUE)
+vx1 <- oinc[1L]; vx2 <- oinc[2L]
+vx3 <- odec[2L]; vx4 <- odec[1L]
+# let's visualize these four vertices on the mesh
+open3d(windowRect = 50 + c(0, 0, 512, 512))
+view3d(0, -25, zoom = 0.8)
+shade3d(rmesh, color = "deeppink4", polygon_offset = 1)
+shade3d(bndry, lwd = 3)
+points3d(rbind(vs[vx1, ]), col = "red",    size = 12)
+points3d(rbind(vs[vx2, ]), col = "green",  size = 12)
+points3d(rbind(vs[vx3, ]), col = "blue",   size = 12)
+points3d(rbind(vs[vx4, ]), col = "yellow", size = 12)
+```
+
+![](./figures/Enneper_fourCorners.png)
+
+Now let's compute the DCP parameterization with these given corners, 
+let's make the checkerboard and let's plot the result:
+```{r, eval=FALSE}
+# compute the DCP parameterization with the four given corners ####
+UV <- mesh$parameterization(method = "DCP", corners = c(vx1, vx2, vx3, vx4))
+# make square checkerboard with 5 squares x 5 squares ####
+checkerboard <- ifelse(
+  (floor(5 * UV[, 1L]) %% 2) == (floor(5 * UV[, 2L]) %% 2), 
+  "gold", "magenta4"
+)
+# add checkerboard colors to the 'rgl' mesh ####
+rmesh[["material"]] <- list("color" = checkerboard)
+# plot ####
+open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.8)
+bg3d("#363940")
+shade3d(rmesh, meshColor = "vertices", polygon_offset = 1)
+shade3d(bndry, lwd = 4)
+```
+
+![](./figures/EnneperCheckerboard_FourCorners.gif)
+
+Now you see what I mean by "align". It is quite better, no?
+
+
+### Second method: ARAP parameterization

inst/essais/essai-param-fourCorners-Enneper.R

@@ -19,8 +19,11 @@ rmesh <- parametricMesh(
 
 bndry <- getBoundary3d(rmesh, sorted = TRUE, color = "black")
 
+open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.8)
+par3d(userMatrix = um)
 shade3d(rmesh, col = "deeppink4")
-bbox3d()
+shade3d(bndry, lwd = 4)
+snapshot3d("EnneperOrderTwo.png", webshot = FALSE)
 
 # convert to CGAL mesh ####
 mesh <- cgalMesh$new(rmesh)
@@ -33,6 +36,28 @@ summary(edges[["length"]])
 mesh$isotropicRemeshing(0.008, iterations = 3, relaxSteps = 2)
 
 # compute mesh parameterization ####
+# without the four corners
+UV <- mesh$parameterization(method = "DCP")
+# the UV-space is the square [0,1]x[0,1]
+# make square checkerboard with 5 squares x 5 squares ####
+checkerboard <- ifelse(
+  (floor(5 * UV[, 1L]) %% 2) == (floor(5 * UV[, 2L]) %% 2), 
+  "gold", "magenta4"
+)
+# add normals, convert to 'rgl' mesh, and add colors ####
+mesh$computeNormals()
+rmesh <- mesh$getMesh()
+rmesh[["material"]] <- list("color" = checkerboard)
+# plot ####
+library(rgl)
+open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.8)
+bg3d("#363940")
+par3d(userMatrix = um)
+shade3d(rmesh, meshColor = "vertices", polygon_offset = 1)
+shade3d(b, lwd = 4)
+snapshot3d(sprintf("EnneperCheckerboard_NoCorners.png"), webshot = FALSE)
+
+# now we will fix four corners ####
 # first, find the four corners
 vs <- mesh$getVertices()
 vsx <- vs[, 1L]
@@ -47,9 +72,9 @@ open3d(windowRect = 50 + c(0, 0, 512, 512))
 view3d(0, -25, zoom = 0.8)
 shade3d(rmesh, color = "deeppink4", polygon_offset = 1)
 shade3d(bndry, lwd = 3)
-points3d(rbind(vs[v1, ]), col = "red", size = 12)
-points3d(rbind(vs[v2, ]), col = "green", size = 12)
-points3d(rbind(vs[v3, ]), col = "blue", size = 12)
+points3d(rbind(vs[v1, ]), col = "red",    size = 12)
+points3d(rbind(vs[v2, ]), col = "green",  size = 12)
+points3d(rbind(vs[v3, ]), col = "blue",   size = 12)
 points3d(rbind(vs[v4, ]), col = "yellow", size = 12)
 snapshot3d("Enneper_fourCorners.png", webshot = FALSE)