square checkerboard for Enneper with four corners and ARAP

stla · Oct 7, 2023 · 9d4fdbd · 9d4fdbd
1 parent f45a4c6
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diff --git a/inst/essais/Enneper.R b/inst/essais/Enneper.R
@@ -1,7 +1,7 @@
 library(cgalMeshes)
 library(rgl)
 
-n <- 3
+n <- 2
 Enneper <- function(phi, r) {
   rbind(
     r*cos(phi) - r^(2*n-1)*cos((2*n-1)*phi)/(2*n-1),
@@ -11,7 +11,9 @@ Enneper <- function(phi, r) {
 }
 
 rmesh <- parametricMesh(
-  Enneper, urange = c(0, 2*pi), vrange = c(0, 1.3),
-  periodic = c(TRUE, FALSE), nu = 200, nv = 100
+  Enneper, urange = c(0, 2*pi), vrange = c(0, 1.2),
+  periodic = c(TRUE, FALSE), nu = 200, nv = 100, clean = TRUE
 )
-rmesh <- Rvcg::vcgClean(rmesh, sel = 0)
+#rmesh <- Rvcg::vcgClean(rmesh, sel = 0)
+
+shade3d(rmesh)
diff --git a/inst/essais/essai-param-ARAP-Enneper.R b/inst/essais/essai-param-ARAP-Enneper.R
@@ -0,0 +1,156 @@
+setwd("C:/SL/MyPackages/cgalMeshes/inst/trash")
+
+library(cgalMeshes)
+library(rgl)
+
+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
+  )
+}
+
+rmesh <- parametricMesh(
+  Enneper, urange = c(0, 2*pi), vrange = c(0, 1.2),
+  periodic = c(TRUE, FALSE), nu = 512L, nv = 128L, clean = TRUE
+)
+
+shade3d(rmesh, col = "deeppink4")
+bbox3d()
+
+# 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 has regular lines ####
+mesh$isotropicRemeshing(0.008, iterations = 3, relaxSteps = 2)
+
+# compute mesh parameterization ####
+UV <- mesh$parameterization(method = "ARAP", lambda = 1000)
+
+png("EnneperARAP_UVspace.png", width = 384, height = 384)
+opar <- par(mar = c(4, 4, 0, 0))
+plot(UV, asp = 1, pch = 19, xlab = "u", ylab = "v", axes = FALSE)
+axis(1, at = seq(-3.5, 1.5, by = 0.5))
+axis(2, at = seq(-1.5, 2.5, by = 0.5))
+par(opar)
+dev.off()
+
+
+vs <- mesh$getVertices()
+vsz <- vs[, 3L]
+odec <- order(vsz, decreasing = TRUE)
+vz1 <- odec[1L]
+vz2 <- odec[2L]
+
+uv1 <- UV[vz1, ]
+uv2 <- UV[vz2, ]
+alpha <- atan2(uv2[1L]-uv1[1L], uv2[2L]-uv1[2L])
+
+rotation <- function(alpha, uv) {
+  t(rbind(
+    c(cos(alpha), -sin(alpha)),
+    c(sin(alpha),  cos(alpha))
+  ) %*% t(uv))
+}
+
+UVrot <- rotation(alpha, UV)
+Urot <- UVrot[, 1L]
+Vrot <- UVrot[, 2L]
+
+png("EnneperARAP_UVspaceWithTheTwoExtremeZPoints.png", width = 384, height = 384)
+opar <- par(mar = c(4, 4, 0, 0))
+plot(UV, asp = 1, pch = ".", xlab = "u", ylab = "v", axes = FALSE)
+axis(1, at = seq(-3.5, 1.5, by = 0.5))
+axis(2, at = seq(-1.5, 2.5, by = 0.5))
+points(UV[c(v1,v2),], col = c("red", "green"), pch = 19, cex = 2)
+par(opar)
+dev.off()
+
+
+# ARAP checkerboard ####
+# U <- UV[, 1L]
+# V <- UV[, 2L]
+# Un <- (U - min(U)) / (max(U) - min(U))
+# Vn <- (V - min(V)) / (max(V) - min(V))
+# checkerboard <- ifelse(
+#   (floor(5 * U) %% 2) == (floor(5 * V) %% 2), 
+#   "yellow", "navy"
+# )
+
+Un <- (Urot - min(Urot)) / (max(Urot) - min(Urot))
+Vn <- (Vrot - min(Vrot)) / (max(Vrot) - min(Vrot))
+checkerboard <- ifelse( # c'est ça le bon
+  (floor(5 * Un) %% 2) == (floor(5 * Vn) %% 2), 
+  "yellow", "navy"
+)
+
+png("Enneper_UVcheckerboard_ARAP.png", width = 384, height = 384)
+opar <- par(mar = c(4, 4, 0, 0))
+plot(
+  U, V, asp = 1, pch = ".", xlab = "u", ylab = "v", 
+  axes = FALSE, col = checkerboard
+)
+axis(1, at = seq(-3.5, 1.5, by = 0.5))
+axis(2, at = seq(-1.5, 2.5, by = 0.5))
+par(opar)
+dev.off()
+
+# add normals, convert to 'rgl' mesh, and add colors ####
+mesh$computeNormals()
+rmesh <- mesh$getMesh()
+rmesh[["material"]] <- list("color" = checkerboard)
+b <- getBoundary3d(rmesh, sorted = TRUE, color = "black")
+
+# plot ####
+library(rgl)
+open3d(windowRect = 50 + c(0, 0, 512, 512), zoom = 0.8)
+bg3d("#363940")
+par3d(userMatrix = um)
+shade3d(rmesh, meshColor = "vertices", polygon_offset = 1)
+shade3d(b, lwd = 4)
+
+snapshot3d(sprintf("EnneperWithCheckerboard_ARAP.png"), webshot = FALSE)
+saveRDS(um, "userMatrix.rds")
+
+# animation ####
+M <- par3d("userMatrix")
+movie3d(
+  par3dinterp(
+    time = seq(0, 1, len = 9),
+    userMatrix = list(
+      M,
+      rotate3d(M, pi, 1, 0, 0),
+      rotate3d(M, pi, 1, 1, 0),
+      rotate3d(M, pi, 1, 1, 1),
+      rotate3d(M, pi, 0, 1, 1),
+      rotate3d(M, pi, 0, 1, 0),
+      rotate3d(M, pi, 1, 0, 1),
+      rotate3d(M, pi, 0, 0, 1),
+      M
+    )
+  ),
+  fps = 100,
+  duration = 1,
+  dir = ".",
+  movie = "zzpic",
+  convert = FALSE, webshot = FALSE
+)
+
+# mount animation
+library(gifski)
+gifski(
+  png_files = Sys.glob("zzpic*.png"),
+  gif_file = "TennisBallWithCheckerboard_FourCorners.gif",
+  width = 512,
+  height = 512,
+  delay = 1/8
+)
+file.remove(Sys.glob("zzpic*.png"))
+
+
diff --git a/inst/essais/essai-param-fourCorners-Enneper.R b/inst/essais/essai-param-fourCorners-Enneper.R
@@ -0,0 +1,112 @@
+setwd("C:/SL/MyPackages/cgalMeshes/inst/trash")
+
+library(cgalMeshes)
+library(rgl)
+
+n <- 2
+Enneper <- function(phi, r) {
+  rbind(
+    r*cos(phi) - r^(2*n-1)*cos((2*n-1)*phi)/(2*n-1),
+    r*sin(phi) + r^(2*n-1)*sin((2*n-1)*phi)/(2*n-1),
+    2*r^n*cos(n*phi)/n
+  )
+}
+
+rmesh <- parametricMesh(
+  Enneper, urange = c(0, 2*pi), vrange = c(0, 1.2),
+  periodic = c(TRUE, FALSE), nu = 512L, nv = 128L, clean = TRUE
+)
+
+shade3d(rmesh, col = "deeppink4")
+bbox3d()
+
+# convert to CGAL mesh ####
+mesh <- cgalMesh$new(rmesh)
+
+# take a look at the edge lengths
+edges <- mesh$getEdges()
+summary(edges[["length"]])
+
+# add vertices in order that the checkeboard has regular lines ####
+mesh$isotropicRemeshing(0.008, iterations = 3, relaxSteps = 2)
+
+# compute mesh parameterization ####
+# first, find the four corners
+vs <- mesh$getVertices()
+vsx <- vs[, 1L]
+oinc <- order(vsx)
+odec <- order(vsx, decreasing = TRUE)
+v1 <- oinc[1L]
+v2 <- oinc[2L]
+v3 <- odec[2L]
+v4 <- odec[1L]
+
+open3d()
+view3d(0, 0)
+shade3d(rmesh)
+points3d(rbind(vs[v1, ]), col = "red", size = 12)
+points3d(rbind(vs[v2, ]), col = "green", size = 12)
+points3d(rbind(vs[v3, ]), col = "blue", size = 12)
+points3d(rbind(vs[v4, ]), col = "black", size = 12)
+
+UV <- mesh$parameterization(method = "DCP", corners = c(v1, v2, v3, v4))
+
+# square checkerboard ####
+checkerboard <- ifelse(
+  (floor(5 * UV[, 1L]) %% 2) == (floor(5 * UV[, 2L]) %% 2), 
+  "yellow", "navy"
+)
+
+# add normals, convert to 'rgl' mesh, and add colors ####
+mesh$computeNormals()
+rmesh <- mesh$getMesh()
+rmesh[["material"]] <- list("color" = checkerboard)
+b <- getBoundary3d(rmesh, sorted = TRUE, color = "black")
+
+# 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("EnneperWithCheckerboard_FourCorners.png"), webshot = FALSE)
+saveRDS(um, "userMatrix.rds")
+
+# animation ####
+M <- par3d("userMatrix")
+movie3d(
+  par3dinterp(
+    time = seq(0, 1, len = 9),
+    userMatrix = list(
+      M,
+      rotate3d(M, pi, 1, 0, 0),
+      rotate3d(M, pi, 1, 1, 0),
+      rotate3d(M, pi, 1, 1, 1),
+      rotate3d(M, pi, 0, 1, 1),
+      rotate3d(M, pi, 0, 1, 0),
+      rotate3d(M, pi, 1, 0, 1),
+      rotate3d(M, pi, 0, 0, 1),
+      M
+    )
+  ),
+  fps = 100,
+  duration = 1,
+  dir = ".",
+  movie = "zzpic",
+  convert = FALSE, webshot = FALSE
+)
+
+# mount animation
+library(gifski)
+gifski(
+  png_files = Sys.glob("zzpic*.png"),
+  gif_file = "TennisBallWithCheckerboard_FourCorners.gif",
+  width = 512,
+  height = 512,
+  delay = 1/8
+)
+file.remove(Sys.glob("zzpic*.png"))
+
+
diff --git a/inst/essais/essai_conformalTorus.R b/inst/essais/essai_conformalTorus.R
@@ -149,3 +149,61 @@ tube <- addNormals(
 shade3d(tube, color = "black")
 
 snapshot3d("Villarceau.png", webshot = FALSE)
+
+
+
+################################################################################
+
+f <- function(beta, theta0, phi = pi/4) {
+  d <- (1 - sin(beta) * sin(phi))
+  rbind(
+    cos(theta0 + beta) * cos(phi) / d,
+    sin(theta0 + beta) * cos(phi) / d,
+    cos(beta) * sin(phi) / d
+  ) 
+}
+
+mesh <- parametricMesh(
+  f, c(0, 2*pi), c(0, 2*pi), c(TRUE, TRUE),
+  nu = 256, nv = 256
+)
+
+f <- function(u, v) { # clifford
+  rbind(
+    cospi(u) / (sqrt(2) - sinpi(v)),
+    sinpi(u) / (sqrt(2) - sinpi(v)),
+    cospi(v) / (sqrt(2) - sinpi(v))
+  ) 
+}
+
+N <- 256
+
+mesh <- parametricMesh(
+  f, c(0, 2), c(0, 2), c(TRUE, TRUE),
+  nu = 256, nv = 256
+)
+
+x_ <- seq(0, 1, length.out = N)
+UV <- as.matrix(
+  expand.grid(U = x_, V = x_)
+)
+
+rotation <- function(alpha, uv) {
+  t(rbind(
+    c(cos(alpha), -sin(alpha)),
+    c(sin(alpha),  cos(alpha))
+  ) %*% t(uv))
+}
+
+UVrot <- rotation(pi/4, UV)
+
+K <- 4*sqrt(2)
+rotatedCheckerboardColors <- ifelse(
+  (floor(K*UVrot[, 1L]) %% 2) == (floor(K*UVrot[, 2L]) %% 2),
+  "yellow", "navy"
+)
+
+mesh$material <- list(color = rotatedCheckerboardColors)
+open3d(windowRect = 50 + c(0, 0, 512, 512))
+view3d(-15, -25, zoom = 0.7)
+shade3d(mesh, polygon_offset = 1) 
diff --git a/vignettes/parameterizations.Rmd b/vignettes/parameterizations.Rmd
@@ -116,7 +116,7 @@ checkerboard: the lengths are not preserved, the angles are not preserved.
 
 A parameterization which preserves the angles is called a *conformal* 
 parameterization. Let's see an example. I know a conformal parameterization 
-of a torus:
+of the torus:
 
 ```{r conformalTorus}
 # the conformal torus mesh
@@ -171,6 +171,7 @@ rotatedCheckerboardColors <- ifelse(
 ```
 
 ```{r plotRotatedCheckerboard, eval=FALSE}
+# plot the rotated checkerboard
 opar <- par(mar = c(0, 0, 0, 0))
 plot(
   UV, col = rotatedCheckerboardColors, asp = 1, pch = ".", 
@@ -181,6 +182,8 @@ par(opar)
 
 ![](rotatedCheckerboard.png)
 
+Now we assign the colors to the vertices of the torus mesh, and we plot.
+
 ```{r}
 # assign the rotated checkerboard colors to the torus mesh
 TorusMesh[["material"]] <- list("color" = rotatedCheckerboardColors)
@@ -193,7 +196,7 @@ view3d(0, -40, zoom = 0.7)
 shade3d(TorusMesh, meshColor = "vertices")
 ```
 
-![](conformalTorus.png)
+![](conformalTorus.gif)
 
 The parameterization is conformal.
 It is not very obvious to see, but the right angles are not distorted 
@@ -202,7 +205,7 @@ In fact, the boundaries of the yellow and blue squares form the *Villarceau
 circles* of the torus, shown in black on the following picture, and which are 
 known to meet at right angles.
 
-
+![](Villarceau.png)