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library(jacobi) | ||
library(RcppColors) | ||
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# the modified Cayley transformation | ||
Phi <- function(z) (1i*z + 1) / (z + 1i) | ||
PhiInv <- function(z) { | ||
1i + (2i*z) / (1i - z) | ||
} | ||
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# background color | ||
bkgcol <- "#ffffff" | ||
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# make the color mapping | ||
f <- function(x, y) { | ||
z <- complex(real = x, imaginary = y) | ||
tau <- PhiInv(z) | ||
ifelse( | ||
Mod(z) >= 1, | ||
NA_complex_, | ||
EisensteinE(4L, exp(1i*pi*tau)) | ||
) | ||
} | ||
x <- y <- seq(-1, 1, length.out = 512L) | ||
S <- outer(x, y, Vectorize(f)) | ||
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img <- colorMap7(Im(S)) | ||
img <- colorMap5(S) | ||
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opar <- par(mar = c(0, 0, 0, 0)) | ||
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, | ||
axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA) | ||
rasterImage(img, 0, 0, 1, 1) | ||
par(opar) | ||
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############################################################################ | ||
n <- 2048L | ||
x <- y <- seq(0.0001, 0.9999, length.out = n) | ||
Grid <- transform( | ||
expand.grid(X = x, Y = y), | ||
Z = complex(real = X, imaginary = Y) | ||
) | ||
Tau <- square2H(Grid$Z) | ||
Q <- exp(1i*pi*Tau) | ||
K <- vapply(Q, function(q) EisensteinE(6L, q), complex(1L)) | ||
dim(K) <- c(n, n) | ||
# plot | ||
if(require("RcppColors")) { | ||
img <- colorMap5(K) | ||
} else { | ||
img <- as.raster((Arg(K) + pi)/(2*pi)) | ||
} | ||
opar <- par(mar = c(0, 0, 0, 0)) | ||
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, | ||
axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA) | ||
rasterImage(img, 0, 0, 1, 1) | ||
par(opar) | ||
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# animation #### | ||
# Möbius transformation of order 3 | ||
Mob <- function(z, t) { | ||
a <- pi*t/3 | ||
((sqrt(3)*cos(a) - sin(a)) * z - 2*sin(a))/ | ||
(2*sin(a) * z + sqrt(3)*cos(a) + sin(a)) | ||
} | ||
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# smooth stair function | ||
xmsinx <- function(x) x - sin(x) | ||
s <- function(x) { | ||
(xmsinx(xmsinx(2*x)) + 4*pi) / (2*pi) | ||
} | ||
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# frames | ||
t_ <- seq(-2*pi, pi, length.out = 161)[-1L] | ||
for(i in seq_along(t_)) { | ||
print(i) | ||
img <- colorMap5(Mob(K, s(t_[i])))#, bkgcolor = bkgcol) | ||
#img <- permuteRGB(img, "brg") | ||
svg("x.svg", width = 16, height = 16) | ||
opar <- par(mar = c(0, 0, 0, 0)) | ||
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, | ||
axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA) | ||
rasterImage(img, 0, 0, 1, 1) | ||
par(opar) | ||
dev.off() | ||
rsvg::rsvg_png( | ||
"x.svg", file = sprintf("zzpic%03d.png", i), | ||
width = 512, height = 512 | ||
) | ||
} | ||
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# mount animation | ||
library(gifski) | ||
pngs <- Sys.glob("zzpic*.png") | ||
gifski( | ||
pngs, | ||
"Eisenstein6OnSquare.gif", | ||
width = 512, height = 512, | ||
delay = 1/10 | ||
) | ||
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file.remove(pngs) |
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@@ -0,0 +1,35 @@ | ||
library(jacobi) | ||
library(RcppColors) | ||
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||
# the modified Cayley transformation | ||
Phi <- function(z) (1i*z + 1) / (z + 1i) | ||
PhiInv <- function(z) { | ||
1i + (2i*z) / (1i - z) | ||
} | ||
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# background color | ||
bkgcol <- "#ffffff" | ||
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# make the color mapping | ||
f <- function(x, y) { | ||
z <- complex(real = x, imaginary = y) | ||
tau <- PhiInv(z) | ||
ifelse( | ||
Mod(z) >= 1, | ||
NA_complex_, | ||
eta(tau) | ||
) | ||
} | ||
x <- y <- seq(-1, 1, length.out = 512L) | ||
S <- outer(x, y, Vectorize(f)) | ||
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img <- colorMap5(S^24) | ||
img <- permuteRGB(img, "gbr") | ||
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opar <- par(mar = c(0, 0, 0, 0)) | ||
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, | ||
axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA) | ||
rasterImage(img, 0, 0, 1, 1) | ||
par(opar) | ||
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library(jacobi) | ||
library(RcppColors) | ||
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# the conformal map | ||
Phi <- function(z) { | ||
-jtheta2(z, 2i) / jtheta3(z, 2i) | ||
} | ||
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# background color | ||
bkgcol <- "#ffffff" | ||
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# make the color mapping | ||
f <- function(x, y) { | ||
z <- complex(real = x, imaginary = y) | ||
ifelse( | ||
x %in% c(0, 1) || y %in% c(0,1), | ||
NA_complex_, | ||
jacobi:::jtheta3(z,Phi(z*pi)) | ||
) | ||
} | ||
x <- y <- seq(0, 1, length.out = 256L) | ||
L <- outer(x, y, Vectorize(f)) | ||
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img <- colorMap5(1/L) | ||
img <- permuteRGB(img, "brg") | ||
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opar <- par(mar = c(0, 0, 0, 0)) | ||
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, | ||
axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA) | ||
rasterImage(img, 0, 0, 1, 1) | ||
par(opar) | ||
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# animation #### | ||
# Möbius transformation of order 3 | ||
Mob <- function(z, t) { | ||
a <- pi*t/3 | ||
((sqrt(3)*cos(a) - sin(a)) * z - 2*sin(a))/ | ||
(2*sin(a) * z + sqrt(3)*cos(a) + sin(a)) | ||
} | ||
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# smooth stair function | ||
xmsinx <- function(x) x - sin(x) | ||
s <- function(x) { | ||
(xmsinx(xmsinx(3*x)) + 6*pi) / (4*pi) | ||
} | ||
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# frames | ||
t_ <- seq(-2*pi, 2*pi, length.out = 161)[-1L] | ||
#t_ <- seq(0, 3, length.out = 61L)[-1L] | ||
for(i in seq_along(t_)) { | ||
print(i) | ||
img <- colorMap5(Mob(L, s(t_[i])))#, bkgcolor = bkgcol) | ||
img <- permuteRGB(img, "brg") | ||
svg("x.svg", width = 16, height = 16) | ||
opar <- par(mar = c(0, 0, 0, 0)) | ||
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, | ||
axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA) | ||
rasterImage(img, 0, 0, 1, 1) | ||
par(opar) | ||
dev.off() | ||
rsvg::rsvg_png( | ||
"x.svg", file = sprintf("zzpic%03d.png", i), | ||
width = 512, height = 512 | ||
) | ||
} | ||
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# mount animation | ||
library(gifski) | ||
pngs <- Sys.glob("zzpic*.png") | ||
gifski( | ||
pngs, | ||
"lambdaOnSquare.gif", | ||
width = 512, height = 512, | ||
delay = 1/11 | ||
) | ||
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file.remove(pngs) |
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@@ -0,0 +1,40 @@ | ||
library(jacobi) | ||
library(RcppColors) | ||
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# the modified Cayley transformation | ||
Phi <- function(z) (1i*z + 1) / (z + 1i) | ||
disk2H <- function(z) { | ||
1i + (2i*z) / (1i - z) | ||
} | ||
disk2H <- function(z) { | ||
1i * (1-z)/(1+z) | ||
} | ||
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# background color | ||
bkgcol <- "#ffffff" | ||
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# map the disk to H and calculate kleinj | ||
f <- function(x, y) { | ||
z <- complex(real = x, imaginary = y) | ||
K <- rep(NA_complex_, length(x)) | ||
inDisk <- Mod(z) < 1 | ||
K[inDisk] <- kleinj(disk2H(z[inDisk])) | ||
K | ||
} | ||
n <- 1024L | ||
x <- y <- seq(-1, 1, length.out = n) | ||
Grid <- expand.grid(X = x, Y = y) | ||
K <- f(Grid$X, Grid$Y) | ||
dim(K) <- c(n, n) | ||
# plot | ||
if(require("RcppColors")) { | ||
img <- colorMap5(K) | ||
} else { | ||
img <- as.raster(1 - abs(Im(K))/Mod(K)) | ||
} | ||
opar <- par(mar = c(0, 0, 0, 0)) | ||
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, | ||
axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA) | ||
rasterImage(img, 0, 0, 1, 1) | ||
par(opar) | ||
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---|---|---|
@@ -0,0 +1,109 @@ | ||
library(jacobi) | ||
library(RcppColors) | ||
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# the modified Cayley transformation | ||
Phi <- function(z) (1i*z + 1) / (z + 1i) | ||
PhiInv <- function(z) { | ||
1i + (2i*z) / (1i - z) | ||
} | ||
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# background color | ||
bkgcol <- "#ffffff" | ||
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|
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# make the color mapping | ||
f <- function(x, y) { | ||
z <- complex(real = x, imaginary = y) | ||
tau <- PhiInv(z) | ||
ifelse( | ||
Mod(z) >= 1, | ||
NA_complex_, | ||
wp(tau/2, omega=c(0.5, 0.5*tau)) | ||
) | ||
} | ||
x <- y <- seq(-1, 1, length.out = 1024L) | ||
W <- outer(x, y, Vectorize(f)) | ||
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img <- colorMap5(W) # 7, 14 is nice as well (for Im(W)^2 / Mod(W)^2) | ||
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opar <- par(mar = c(0, 0, 0, 0)) | ||
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, | ||
axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA) | ||
rasterImage(img, 0, 0, 1, 1) | ||
par(opar) | ||
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svg("x.svg", width = 16, height = 16) | ||
opar <- par(mar = c(0, 0, 0, 0)) | ||
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, | ||
axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA) | ||
rasterImage(img, 0, 0, 1, 1) | ||
par(opar) | ||
dev.off() | ||
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rsvg::rsvg_png( | ||
"x.svg", "wp_Cayley_cm14.png", width = 512, height = 512 | ||
) | ||
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# animation Möbius #### | ||
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# Möbius transformation of order 2 | ||
Mob <- function(z, gamma = -0.5 + 0.5i, t){ | ||
h <- sqrt(1-Mod(gamma)^2) | ||
d2 <- h^t * (cos(t*pi/2) + 1i*sin(t*pi/2)) | ||
d1 <- Conj(d2) | ||
a <- Re(d1) - 1i*Im(d1)/h | ||
b <- gamma * Im(d2)/h | ||
c <- Conj(b) | ||
d <- Conj(a) | ||
(a*z + b) / (c*z + d) | ||
} | ||
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f <- function(x, y, t) { | ||
z <- complex(real = x, imaginary = y) | ||
z <- Mob(z, t = t) | ||
tau <- PhiInv(z) | ||
ifelse( | ||
Mod(z) >= 1, | ||
NA_complex_, | ||
wp(tau/2, omega=c(0.5, 0.5*tau)) | ||
) | ||
} | ||
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s <- function(x) x | ||
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t_ <- seq(0, 2, length.out = 61)[-1L] | ||
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for(i in seq_along(t_)) { | ||
print(i) | ||
#MobW <- Mob(W, t = s(t_[i])) # c pas ça qu'il faut faire... il faut refaire un outer | ||
W <- outer(x, y, Vectorize(f), t = s(t_[i])) | ||
img <- colorMap14(Im(W)^2 / Mod(W)^2)#, bkgcolor = bkgcol) | ||
svg("x.svg", width = 16, height = 16) | ||
opar <- par(mar = c(0, 0, 0, 0)) | ||
plot(NULL, xlim = c(0, 1), ylim = c(0, 1), asp = 1, | ||
axes = FALSE, xaxs = "i", yaxs = "i", xlab = NA, ylab = NA) | ||
rasterImage(img, 0, 0, 1, 1) | ||
par(opar) | ||
dev.off() | ||
rsvg::rsvg_png( | ||
"x.svg", file = sprintf("zzpic%03d.png", i), | ||
width = 512, height = 512 | ||
) | ||
} | ||
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# mount animation | ||
library(gifski) | ||
pngs <- Sys.glob("zzpic*.png") | ||
gifski( | ||
pngs, | ||
"wp_Cayley.gif", | ||
width = 512, height = 512, | ||
delay = 1/8 | ||
) | ||
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file.remove(pngs) | ||
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