3D plot of a Klein Bottle
21 Jun 2015The Klein bottle was discovered in 1882 by Felix Klein and since then has joined the gallery of popular mathematical shapes.
The Klein bottle is a one-sided closed surface named after Klein. We note however this is not a continuous surface in 3-space as the surface cannot go through itself without a discontinuity.
#install.packages('animation', repos = 'http://rforge.net', type = 'source')
#install.packages("plot3D")
library(plot3D)
library(animation)
saveGIF({
par(mai = c(0.1,0.1,0.1,0.1))
r = 3
for(i in 1:100){
X <- seq(0, 2*pi, length.out = 200)
#Y <- seq(-15, 6, length.out = 100)
Y <- seq(0, 2*pi, length.out = 200)
M <- mesh(X, Y)
u <- M$x
v <- M$y
# x, y and z grids
# x <- (1.16 ^ v) * cos(v) * (1 + cos(u))
# y <- (-1.16 ^ v) * sin(v) * (1 + cos(u))
# z <- (-2 * 1.16 ^ v) * (1 + sin(u))
###### Klein bottle ##########
x <- (r + cos(u/2) * sin(v) - sin(u/2) * sin(2*v)) * cos(u)
y <- (r + cos(u/2) * sin(v) - sin(u/2) * sin(2*v)) * sin(u)
z <- sin(u/2) * sin(v) + cos(u/2) * sin(2*v)
##################################
# full colored image
surf3D(x, y, z, colvar = z,
col = ramp.col(col = c("red", "red", "orange"), n = 100),
colkey = FALSE, shade = 0.5, expand = 1.2, box = FALSE,
phi = 35, theta = i, lighting = TRUE, ltheta = 560,
lphi = -i)
}
}, interval = 0.1, ani.width = 550, ani.height = 350)## [1] TRUEHere is the result from the code above:
References
[Klein Bottle] (https://www.math.hmc.edu/funfacts/ffiles/30002.7.shtml)
[http://genedan.com/no-39-exploring-sage-as-an-alternative-to-mathematica/] (http://genedan.com/no-39-exploring-sage-as-an-alternative-to-mathematica/)
[Imaging maths - Inside the Klein bottle] (https://plus.maths.org/content/imaging-maths-inside-klein-bottle)
[Fun with surf3D function] (http://alstatr.blogspot.pt/2014/02/r-fun-with-surf3d-function.html)