Package 'gyro'

Changes of sign

Description

Sometimes, the coordinates of the vertices of a polyhedron are given with changes of sign (with a symbol +/-). This function performs the changes of sign.

Usage

changesOfSign(M, changes = "all")

Arguments

M	a numeric matrix of coordinates of some points (one point per row)
changes	either the indices of the columns of M where the changes of sign must be done, or "all" to select all the indices

Value

A numeric matrix, M transformed by the changes of sign.

Examples

library(gyro)
library(rgl)
## ~~ rhombicosidodecahedron ~~##
phi <- (1 + sqrt(5)) / 2
vs1 <- rbind(
  c(1, 1, phi^3),
  c(phi^2, phi, 2 * phi),
  c(2 + phi, 0, phi^2)
)
vs2 <- rbind(vs1, vs1[, c(2, 3, 1)], vs1[, c(3, 1, 2)]) # even permutations
vs <- changesOfSign(vs2)
open3d(windowRect = c(50, 50, 562, 562), zoom = 0.65)
plotGyrohull3d(vs)

---

Point on a gyroline

Description

Point of coordinate t on the gyroline passing through two given points A and B. This is A for t=0 and this is B for t=1. For t=1/2 this is the gyromidpoint of the gyrosegment joining A and B.

Usage

gyroABt(A, B, t, s = 1, model = "U")

Arguments

A, B	two distinct points
t	a number
s	positive number, the radius of the Poincaré ball if model="M", otherwise, if model="U", this number defines the hyperbolic curvature
model	the hyperbolic model, either "M" (Möbius model, i.e. Poincaré model) or "U" (Ungar model, i.e. hyperboloid model)

Value

A point.

---

Gyrocentroid

Description

Gyrocenroid of a triangle.

Usage

gyrocentroid(A, B, C, s = 1, model = "U")

Arguments

A, B, C	three distinct points
s	positive number, the radius of the Poincaré ball if model="M", otherwise, if model="U", this number defines the hyperbolic curvature (the smaller, the more curved)
model	the hyperbolic model, either "M" (Möbius model, i.e. Poincaré model) or "U" (Ungar model, i.e. hyperboloid model)

Value

A point, the gyrocentroid of the triangle ABC.

---

Examples of the 'gyro' package

Description

Some examples of hyperbolic polyhedra realized with the 'gyro' package.

Usage

gyrodemos()

Value

No value. The function firstly copies the demo files in a temporary directory. If you use RStudio, the function opens these files. Otherwise it prints a message giving the instructions to access to these files.

Note

The BarthLike file has this name because the figure it generates looks like the Barth sextic (drawing by Patrice Jeener):

---

Gyromidpoint

Description

The gyromidpoint of a gyrosegment.

Usage

gyromidpoint(A, B, s = 1, model = "U")

Arguments

A, B	two distinct points (of the same dimension)
s	positive number, the radius of the Poincaré ball if model="M", otherwise, if model="U", this number defines the hyperbolic curvature
model	the hyperbolic model, either "M" (Möbius model, i.e. Poincaré model) or "U" (Ungar model, i.e. hyperboloid model)

Value

A point, the gyromidpoint of a the gyrosegment joining A and B.

Note

This is the same as gyroABt(A, B, 1/2, s) but the calculation is more efficient.

---

Gyroray

Description

Gyroray given an origin and a point.

Usage

gyroray(O, A, s = 1, tmax = 20, OtoA = TRUE, model = "U", n = 300)

Arguments

O, A	two distinct points (of the same dimension); the point O is the origin of the gyroray
s	positive number, the radius of the Poincaré ball if model="M", otherwise, if model="U", this number defines the hyperbolic curvature
tmax	positive number controlling the length of the gyroray
OtoA	Boolean, whether the gyroray must be directed from O to A or must be the opposite one
model	the hyperbolic model, either "M" (Möbius model, i.e. Poincaré model) or "U" (Ungar model, i.e. hyperboloid model)
n	number of points forming the gyroray

Value

A numeric matrix with n rows. Each row is a point of the gyroray with origin O (the first row) and passing through A or not, according to OtoA.

Examples

library(gyro)
# a 2D example ####
O <- c(1, 2); A <- c(1, 1)
opar <- par(mar = c(2, 2, 2, 0.5))
plot(rbind(O, A), type = "p", pch = 19, xlab = NA, ylab = NA,
     xlim = c(0, 2), ylim = c(0, 3), main = "s = 0.3")
s <- 0.3
ray <- gyroray(O, A, s)
lines(ray, col = "blue", lwd = 2)
text(t(O), expression(italic(O)), pos = 2)
text(t(A), expression(italic(A)), pos = 3)
# opposite gyroray
yar <- gyroray(O, A, s, OtoA = FALSE)
lines(yar, col = "red", lwd = 2)
par(opar)

---

Gyroreflection

Description

Gyroreflection of a point with respect to a gyroline in a 2D gyrospace.

Usage

gyroreflection(A, B, M, s, model = "U")

Arguments

A, B, M	three 2D points
s	the gyroparameter (radius of the Poincaré disk if model="M")
model	the hyperbolic model, either "M" (Möbius model, i.e. Poincaré model) or "U" (Ungar model, i.e. Minkowski model)

Value

A 2D point, the image of M by the reflection with respect to the gyroline passing through A and B.

Examples

library(gyro)
A <- c(1.5, 2); B <- c(2, 1)
opar <- par(mar = c(2, 2, 2, 0.5))
plot(rbind(A, B), type = "p", pch = 19, xlab = NA, ylab = NA,
     xlim = c(0.3, 2), ylim = c(0, 2.5), main = "s = 0.3")
s <- 0.3
seg <- gyrosegment(A, B, s = s, model = "U")
lines(seg, col = "blue", lwd = 2)
text(t(A), expression(italic(A)), pos = 3)
text(t(B), expression(italic(B)), pos = 3)
M <- c(1.3, 1.1)
rM <- gyroreflection(A, B, M, s = s, model = "U")
points(rbind(M, rM), type = "p", pch = 19)
text(t(M), expression(italic(M)), pos = 3)
text(t(rM), expression(italic(r(M))), pos = 3)
par(opar)

---

Gyrosegment

Description

Gyrosegment joining two given points.

Usage

gyrosegment(A, B, s = 1, model = "U", n = 100)

Arguments

A, B	two distinct points (of the same dimension)
s	positive number, the radius of the Poincaré ball if model="M", otherwise, if model="U", this number defines the hyperbolic curvature
model	the hyperbolic model, either "M" (Möbius model, i.e. Poincaré model) or "U" (Ungar model, i.e. hyperboloid model)
n	number of points forming the gyrosegment from A to B

Value

A numeric matrix with n rows. Each row is a point of the gyrosegment from A (the first row) to B (the last row).

Note

The gyrosegment is obtained from gyroABt by varying t from 0 to 1.

Examples

library(gyro)
# a 2D example ####
A <- c(1, 2); B <- c(1, 1)
opar <- par(mfrow = c(1, 2), mar = c(2, 2, 2, 0.5))
plot(rbind(A, B), type = "p", pch = 19, xlab = NA, ylab = NA,
     xlim = c(0, 2), ylim = c(0, 2), main = "s = 0.2")
s <- 0.2
AB <- gyrosegment(A, B, s)
lines(AB, col = "blue", lwd = 2)
text(t(A), expression(italic(A)), pos = 2)
text(t(B), expression(italic(B)), pos = 3)
# this is an hyperbola whose asymptotes meet at the origin
# approximate asymptotes
lines(rbind(c(0, 0), gyroABt(A, B, t = -20, s)), lty = "dashed")
lines(rbind(c(0, 0), gyroABt(A, B, t = 20, s)), lty = "dashed")
# plot the gyromidoint
points(
 rbind(gyromidpoint(A, B, s)),
 type = "p", pch = 19, col = "red"
)
# another one, with a different `s`
plot(rbind(A, B), type = "p", pch = 19, xlab = NA, ylab = NA,
     xlim = c(0, 2), ylim = c(0, 2), main = "s = 0.1")
s <- 0.1
AB <- gyrosegment(A, B, s)
lines(AB, col = "blue", lwd = 2)
text(t(A), expression(italic(A)), pos = 2)
text(t(B), expression(italic(B)), pos = 3)
# approximate asymptotes
lines(rbind(c(0, 0), gyroABt(A, B, t = -20, s)), lty = "dashed")
lines(rbind(c(0, 0), gyroABt(A, B, t = 20, s)), lty = "dashed")
# plot the gyromidoint
points(
 rbind(gyromidpoint(A, B, s)),
 type = "p", pch = 19, col = "red"
)

# a 3D hyperbolic triangle ####
library(rgl)
A <- c(1, 0, 0); B <- c(0, 1, 0); C <- c(0, 0, 1)
s <- 0.3
AB <- gyrosegment(A, B, s)
AC <- gyrosegment(A, C, s)
BC <- gyrosegment(B, C, s)
view3d(30, 30, zoom = 0.75)
lines3d(AB, lwd = 3); lines3d(AC, lwd = 3); lines3d(BC, lwd = 3)

---

Gyrotriangle in 3D space

Description

3D gyrotriangle as a mesh.

Usage

gyrotriangle(
  A,
  B,
  C,
  s = 1,
  model = "U",
  iterations = 5,
  palette = NULL,
  bias = 1,
  interpolate = "linear",
  g = identity
)

Arguments

A, B, C	three distinct 3D points
s	positive number, the radius of the Poincaré ball if model="M", otherwise, if model="U", this number defines the hyperbolic curvature (the smaller, the more curved)
model	the hyperbolic model, either "M" (Möbius model, i.e. Poincaré model) or "U" (Ungar model, i.e. hyperboloid model)
iterations	the gyrotriangle is constructed by iterated subdivisions, this argument is the number of iterations
palette	a vector of colors to decorate the triangle, or NULL if you don't want to use a color palette
bias, interpolate	if palette is not NULL, these arguments are passed to colorRamp
g	a function from [0,1] to [0,1]; if palette is not NULL, this function is applied to the scalars defining the colors (the normalized gyrodistances to the gyrocentroid of the gyrotriangle)

Value

A mesh3d object.

Examples

library(gyro)
library(rgl)
A <- c(1, 0, 0); B <- c(0, 1, 0); C <- c(0, 0, 1)
ABC <- gyrotriangle(A, B, C, s = 0.3)
open3d(windowRect = c(50, 50, 562, 562))
view3d(30, 30, zoom = 0.75)
shade3d(ABC, color = "navy", specular = "cyan")

# using a color palette ####
if(require("trekcolors")) {
  pal <- trek_pal("klingon")
} else {
  pal <- hcl.colors(32L, palette = "Rocket")
}
ABC <- gyrotriangle(
  A, B, C, s = 0.5,
  palette = pal, bias = 1.5, interpolate = "spline"
)
open3d(windowRect = c(50, 50, 562, 562))
view3d(zoom = 0.75)
shade3d(ABC)

# hyperbolic icosahedron ####
library(rgl)
library(Rvcg) # to get the edges with the `vcgGetEdge` function
icosahedron <- icosahedron3d() # mesh with 12 vertices, 20 triangles
vertices <- t(icosahedron$vb[-4, ])
triangles <- t(icosahedron$it)
edges <- as.matrix(vcgGetEdge(icosahedron)[, c("vert1", "vert2")])
s <- 0.3
open3d(windowRect = c(50, 50, 562, 562))
view3d(zoom = 0.75)
for(i in 1:nrow(triangles)){
  triangle <- triangles[i, ]
  A <- vertices[triangle[1], ]
  B <- vertices[triangle[2], ]
  C <- vertices[triangle[3], ]
  gtriangle <- gyrotriangle(A, B, C, s)
  shade3d(gtriangle, color = "midnightblue")
}
for(i in 1:nrow(edges)){
  edge <- edges[i, ]
  A <- vertices[edge[1], ]
  B <- vertices[edge[2], ]
  gtube <- gyrotube(A, B, s, radius = 0.03)
  shade3d(gtube, color = "lemonchiffon")
}
spheres3d(vertices, radius = 0.05, color = "lemonchiffon")

---

Gyrotube (tubular gyrosegment)

Description

Tubular gyrosegment joining two given 3D points.

Usage

gyrotube(A, B, s = 1, model = "U", n = 100, radius, sides = 90, caps = FALSE)

Arguments

A, B	distinct 3D points
s	positive number, the radius of the Poincaré ball if model="M", otherwise, if model="U", this number defines the hyperbolic curvature (higher value, less curved)
model	the hyperbolic model, either "M" (Möbius model, i.e. Poincaré model) or "U" (Ungar model, i.e. hyperboloid model)
n	number of points forming the gyrosegment
radius	radius of the tube around the gyrosegment
sides	number of sides in the polygon cross section
caps	Boolean, whether to put caps on the ends of the tube

Value

A mesh3d object.

Examples

library(gyro)
library(rgl)
A <- c(1, 2, 0); B <- c(1, 1, 0)
tube <- gyrotube(A, B, s = 0.2, radius = 0.02)
shade3d(tube, color = "orangered")

# a 3D hyperbolic triangle ####
library(rgl)
A <- c(1, 0, 0); B <- c(0, 1, 0); C <- c(0, 0, 1)
s <- 0.3
r <- 0.03
AB <- gyrotube(A, B, s, radius = r)
AC <- gyrotube(A, C, s, radius = r)
BC <- gyrotube(B, C, s, radius = r)
view3d(30, 30, zoom = 0.75)
shade3d(AB, color = "gold")
shade3d(AC, color = "gold")
shade3d(BC, color = "gold")
spheres3d(rbind(A, B, C), radius = 0.04, color = "gold")

---

Hyperbolic Delaunay triangulation

Description

Computes the hyperbolic Delaunay triangulation of a set of points.

Usage

hdelaunay(points, model = "M")

Arguments

points	points in the unit disk given as a numeric matrix with two columns
model	the hyperbolic model, either "M" (Möbius model, i.e. Poincaré model) or "U" (Ungar model, i.e. hyperboloid model)

Value

A list with five fields vertices, edges, triangles, ntriangles, and centroids, a matrix giving the gyrocentroids of the triangles. The input points matrix and the output vertices matrix are the same up to the order of the rows if model="M", and if model="U", the points in the output vertices matrix are obtained by isomorphism.

See Also

plotHdelaunay

Examples

library(gyro)
library(uniformly)
set.seed(666)
points <- runif_in_sphere(10L, d = 2)
hdelaunay(points)

---

Hyperbolic reflection

Description

Hyperbolic reflection in the Poincaré disk.

Usage

hreflection(A, B, M)

Arguments

A, B	two points in the Poincaré disk defining the reflection line
M	a point in the Poincaré disk to be reflected

Value

A point in the Poincaré disk, the image of M by the hyperbolic reflection with respect to the line passing through A and B.

Examples

library(gyro)
library(plotrix)
A <- c(0.45, 0.78)
B <- c(0.1, -0.5)
M <- c(0.7, 0)
opar <- par(mar = c(0, 0, 0, 0))
plot(NULL, type = "n", xlim = c(-1, 1), ylim = c(-1, 1), asp = 1,
     axes = FALSE, xlab = NA, ylab = NA)
draw.circle(0, 0, radius = 1, lwd = 2)
lines(gyrosegment(A, B, model = "M"))
points(rbind(A, B), pch = 19)
points(rbind(M), pch = 19, col = "blue")
P <- hreflection(A, B, M)
points(rbind(P), pch = 19, col = "red")
par(opar)

---

Isomorphism from Ungar gyrovector space to Möbius gyrovector space

Description

Isomorphism from the Ungar gyrovector space to the Möbius gyrovector space.

Usage

PhiMU(A, s = 1)

Arguments

A	a point in the Ungar vector space with curvature s
s	a positive number, the hyperbolic curvature of the Ungar vector space

Value

The point of the Poincaré ball of radius s corresponding to A by isomorphism.

---

Isomorphism from Möbius gyrovector space to Ungar gyrovector space

Description

Isomorphism from the Möbius gyrovector space to the Ungar gyrovector space.

Usage

PhiUM(A, s = 1)

Arguments

A	a point whose norm is lower than s
s	positive number, the radius of the Poincaré ball

Value

The point of the Ungar gyrovector space corresponding to A by isomorphism.

---

Plot hyperbolic convex hull

Description

Plot the hyperbolic convex hull of a set of 3D points.

Usage

plotGyrohull3d(
  points,
  s = 1,
  model = "U",
  iterations = 5,
  n = 100,
  edgesAsTubes = TRUE,
  verticesAsSpheres = edgesAsTubes,
  edgesColor = "yellow",
  spheresColor = edgesColor,
  tubesRadius = 0.03,
  spheresRadius = 0.05,
  facesColor = "navy",
  bias = 1,
  interpolate = "linear",
  g = identity
)

Arguments

points	matrix of 3D points, one point per row
s	positive number, the radius of the Poincaré ball if model="M", otherwise, if model="U", this number defines the hyperbolic curvature (the smaller, the more curved)
model	the hyperbolic model, either "M" (Möbius model, i.e. Poincaré model) or "U" (Ungar model, i.e. hyperboloid model)
iterations	argument passed to gyrotriangle
n	argument passed to gyrotube or gyrosegment, the number of points for each edge
edgesAsTubes	Boolean, whether to represent tubular edges
verticesAsSpheres	Boolean, whether to represent the vertices as spheres
edgesColor	a color for the edges
spheresColor	a color for the spheres, if verticesAsSpheres = TRUE
tubesRadius	radius of the tubes, if edgesAsTubes = TRUE
spheresRadius	radius of the spheres, if verticesAsSpheres = TRUE
facesColor	this argument sets the color of the faces; it can be either a single color or a color palette, i.e. a vector of colors; if it is a color palette, it will be passed to the argument palette of gyrotriangle
bias, interpolate, g	these arguments are passed to gyrotriangle in the case when facesColor is a color palette

Value

No value, called for plotting.

Examples

library(gyro)
library(rgl)
# Triangular orthobicopula ####
points <- rbind(
  c(1, -1/sqrt(3), sqrt(8/3)),
  c(1, -1/sqrt(3), -sqrt(8/3)),
  c(-1, -1/sqrt(3), sqrt(8/3)),
  c(-1, -1/sqrt(3), -sqrt(8/3)),
  c(0, 2/sqrt(3), sqrt(8/3)),
  c(0, 2/sqrt(3), -sqrt(8/3)),
  c(1, sqrt(3), 0),
  c(1, -sqrt(3), 0),
  c(-1, sqrt(3), 0),
  c(-1, -sqrt(3), 0),
  c(2, 0, 0),
  c(-2, 0, 0)
)
open3d(windowRect = c(50, 50, 562, 562))
view3d(zoom = 0.7)
plotGyrohull3d(points, s = 0.4)

# a non-convex polyhedron with triangular faces ####
vertices <- rbind(
  c(-2.1806973249, -2.1806973249, -2.1806973249),
  c(-3.5617820682, 0.00000000000, 0.00000000000),
  c(0.00000000000, -3.5617820682, 0.00000000000),
  c(0.00000000000, 0.00000000000, -3.5617820682),
  c(-2.1806973249, -2.1806973249, 2.18069732490),
  c(0.00000000000, 0.00000000000, 3.56178206820),
  c(-2.1806973249, 2.18069732490, -2.1806973249),
  c(0.00000000000, 3.56178206820, 0.00000000000),
  c(-2.1806973249, 2.18069732490, 2.18069732490),
  c(2.18069732490, -2.1806973249, -2.1806973249),
  c(3.56178206820, 0.00000000000, 0.00000000000),
  c(2.18069732490, -2.1806973249, 2.18069732490),
  c(2.18069732490, 2.18069732490, -2.1806973249),
  c(2.18069732490, 2.18069732490, 2.18069732490))
triangles <- 1 + rbind(
  c(3, 2, 0),
  c(0, 1, 3),
  c(2, 1, 0),
  c(4, 2, 5),
  c(5, 1, 4),
  c(4, 1, 2),
  c(6, 7, 3),
  c(3, 1, 6),
  c(6, 1, 7),
  c(5, 7, 8),
  c(8, 1, 5),
  c(7, 1, 8),
  c(9, 2, 3),
  c(3, 10, 9),
  c(9, 10, 2),
  c(5, 2, 11),
  c(11, 10, 5),
  c(2, 10, 11),
  c(3, 7, 12),
  c(12, 10, 3),
  c(7, 10, 12),
  c(13, 7, 5),
  c(5, 10, 13),
  c(13, 10, 7))
edges0 <- do.call(c, lapply(1:nrow(triangles), function(i){
  face <- triangles[i, ]
  list(
    sort(c(face[1], face[2])),
    sort(c(face[1], face[3])),
    sort(c(face[2], face[3]))
  )
}))
edges <- do.call(rbind, edges0)
edges <- edges[!duplicated(edges), ]
s <- 2
library(rgl)
open3d(windowRect = c(50, 50, 1074, 562))
mfrow3d(1, 2)
view3d(zoom = 0.65)
for(i in 1:nrow(triangles)){
  triangle <- triangles[i, ]
  A <- vertices[triangle[1], ]
  B <- vertices[triangle[2], ]
  C <- vertices[triangle[3], ]
  gtriangle <- gyrotriangle(A, B, C, s)
  shade3d(gtriangle, color = "violetred")
}
for(i in 1:nrow(edges)){
  edge <- edges[i, ]
  A <- vertices[edge[1], ]
  B <- vertices[edge[2], ]
  gtube <- gyrotube(A, B, s, radius = 0.06)
  shade3d(gtube, color = "darkviolet")
}
spheres3d(vertices, radius = 0.09, color = "deeppink")
# now plot the hyperbolic convex hull
next3d()
view3d(zoom = 0.65)
plotGyrohull3d(vertices, s)

# an example of color palette ####
if(require("trekcolors")) {
  pal <- trek_pal("lcars_series")
} else {
  pal <- hcl.colors(32L, palette = "Rocket")
}
set.seed(666) # 50 random points on sphere
if(require("uniformly")) {
  points <- runif_on_sphere(50L, d = 3L)
} else {
  points <- matrix(rnorm(50L * 3L), nrow = 50L, ncol = 3L)
  points <- points / sqrt(apply(points, 1L, crossprod))
}
open3d(windowRect = c(50, 50, 562, 562))
plotGyrohull3d(
  points, edgesColor = "brown",
  facesColor = pal, g = function(u) 1-u^2
)

---

Plot hyperbolic mesh

Description

Plot the hyperbolic version of a triangle 3D mesh.

Usage

plotGyroMesh(
  mesh,
  s = 1,
  model = "U",
  iterations = 5,
  n = 100,
  edges = TRUE,
  edgesAsTubes = TRUE,
  edgesColor = "yellow",
  tubesRadius = 0.03,
  verticesAsSpheres = edgesAsTubes,
  spheresColor = edgesColor,
  spheresRadius = 0.05,
  facesColor = "navy",
  bias = 1,
  interpolate = "linear",
  g = identity
)

Arguments

mesh	there are two possibilities for this argument; it can be a triangle rgl mesh (class mesh3d) or a list with (at least) two fields: vertices, a numeric matrix with three columns, and faces, an integer matrix with three columns
s	positive number, the radius of the Poincaré ball if model="M", otherwise, if model="U", this number defines the hyperbolic curvature (the smaller, the more curved)
model	the hyperbolic model, either "M" (Möbius model, i.e. Poincaré model) or "U" (Ungar model, i.e. hyperboloid model)
iterations	argument passed to gyrotriangle
n	argument passed to gyrotube or gyrosegment, the number of points for each edge
edges	Boolean, whether to plot the edges (as tubes or as lines)
edgesAsTubes	Boolean, whether to plot tubular edges; if FALSE, the edges are plotted as lines
edgesColor	a color for the edges
tubesRadius	radius of the tubes, if edgesAsTubes = TRUE
verticesAsSpheres	Boolean, whether to plot the vertices as spheres; if FALSE, the vertices are not plotted
spheresColor	a color for the spheres, if verticesAsSpheres = TRUE
spheresRadius	radius of the spheres, if verticesAsSpheres = TRUE
facesColor	this argument sets the color of the faces; it can be either a single color or a color palette, i.e. a vector of colors; if it is a color palette, it will be passed to the argument palette of gyrotriangle
bias, interpolate, g	these arguments are passed to gyrotriangle in the case when facesColor is a color palette

Value

No value, called for plotting.

Examples

# hyperbolic great stellated dodecahedron
library(gyro)
library(rgl)
GSD <- system.file(
  "extdata", "greatStellatedDodecahedron.ply", package = "gyro"
)
mesh <- Rvcg::vcgPlyRead(GSD, updateNormals = FALSE, clean = FALSE)
open3d(windowRect = c(50, 50, 562, 562), zoom = 0.7)
plotGyroMesh(
  mesh,
  edgesAsTubes = FALSE, edgesColor = "black",
  facesColor = "firebrick1"
)

---

Plot hyperbolic Delaunay triangulation

Description

Plot a hyperbolic Delaunay triangulation obtained with hdelaunay.

Usage

plotHdelaunay(
  hdel,
  vertices = TRUE,
  edges = TRUE,
  circle = TRUE,
  color = "random",
  distinctArgs = list(seedcolors = c("#ff0000", "#00ff00", "#0000ff")),
  randomArgs = list(hue = "random", luminosity = "bright")
)

Arguments

hdel	an output of hdelaunay
vertices	Boolean, whether to plot the vertices
edges	Boolean, whether to plot the edges
circle	Boolean, whether to plot the unit circle; ignored for the Ungar model
color	this argument controls the colors of the triangles; it can be NA for no color, "random" for random colors generated with randomColor, "distinct" for distinct colors generated with createPalette, a single color, a vector of colors (color i attributed to the i-th triangle), or a vectorized function mapping each point in the unit interval to a color
distinctArgs	if color = "distinct", a list of arguments passed to createPalette
randomArgs	if color = "random", a list of arguments passed to randomColor

Value

No returned value, just generates a plot.

Examples

library(gyro)
library(uniformly)
set.seed(666)

points <- runif_in_sphere(35L, d = 2)
hdel <- hdelaunay(points, model = "M")
plotHdelaunay(hdel)

points <- runif_in_sphere(35L, d = 2, r = 0.7)
hdel <- hdelaunay(points, model = "U")
plotHdelaunay(hdel)

# example with colors given by a function ####
library(gyro)
if(require("trekcolors")) {
  pal <- trek_pal("klingon")
} else {
  pal <- hcl.colors(32L, palette = "Rocket")
}

phi <- (1 + sqrt(5)) / 2
theta <- head(seq(0, pi/2, length.out = 11), -1L)
a <- phi^((2*theta/pi)^0.8 - 1)
u <- a * cos(theta)
v <- a * sin(theta)
x <- c(0, u, -v, -u, v)
y <- c(0, v, u, -v, -u)
pts <- cbind(x, y) / 1.03

hdel <- hdelaunay(pts, model = "M")

fcolor <- function(t){
  RGB <- colorRamp(pal)(t)
  rgb(RGB[, 1L], RGB[, 2L], RGB[, 3L], maxColorValue = 255)
}

plotHdelaunay(
  hdel, vertices = FALSE, circle = FALSE, color = fcolor
)

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Hyperbolic tiling

Description

Draw a hyperbolic tiling of the Poincaré disk.

Usage

tiling(n, p, depth = 4, colors = c("navy", "yellow"), circle = TRUE, ...)

Arguments

n, p	two positive integers satisfying 1/n + 1/p < 1/2
depth	positive integer, the number of recursions
colors	two colors to fill the hyperbolic tiling
circle	Boolean, whether to draw the unit circle
...	additional arguments passed to draw.circle

Value

No returned value, just draws the hyperbolic tiling.

Note

The higher value of n, the slower. And of course increasing depth slows down the rendering. The value of p has no influence on the speed.

Examples

library(gyro)
tiling(3, 7, border = "orange")

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