--- title: "Hopf Torus (2/3): the bent equatorial case" author: "Stéphane Laurent" date: "2018-05-01" output: md_document: variant: markdown preserve_yaml: true html_document: keep_md: no prettify: yes linenums: yes prettifycss: twitter-bootstrap tags: R, graphics, rgl highlighter: kate --- --- author: Stéphane Laurent date: '2018-05-01' highlighter: kate linenums: True output: html_document: keep_md: False md_document: preserve_yaml: True variant: markdown prettify: True prettifycss: 'twitter-bootstrap' tags: 'R, graphics, rgl' title: 'Hopf Torus (2/3): the bent equatorial case' --- In this second part, we will see what happens when we map the points lying on a bent equator.  {.r} hopfinverse <- function(q, t){ 1/sqrt(2*(1+q)) * c(q*cos(t)+q*sin(t), sin(t)*(1+q), cos(t)*(1+q), q*sin(t)-q*cos(t)) } stereog <- function(x){ c(x, x, x)/(1-x) }  The sphere with the bent equator will be plotted thanks to the following functions. ![](figures/SphereWithSlopedEquator.png) To rotate the equator to the bent equator, we used the rotation matrix $$R_x = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \alpha & -\sin \alpha \\ 0 & \sin \alpha & \cos \alpha \end{pmatrix}$$  {.r} Rx <- function(alpha) { rbind(c(1, 0, 0), c(0, cos(alpha), -sin(alpha)), c(0, sin(alpha), cos(alpha))) }  Now, let's see the Hopf torus.  {.r} open3d(windowRect=c(50,50,500,500)) view3d(45,45) t_ <- seq(0, 2*pi, len=200) theta_ <- seq(0, 2*pi, len=300) phi <- 0 for(i in seq_along(theta_)){ theta <- theta_[i] rotated <- c(Rx(-pi/8) %*% c(cos(theta)*cos(phi), sin(theta)*cos(phi), sin(phi))) circle4d <- sapply(t_, function(t){ hopfinverse(rotated, t) }) circle3d <- t(apply(circle4d, 2, stereog)) shade3d(cylinder3d(circle3d, radius=0.1), color="purple") }  We get a deformed torus, still made of circles:  