--- 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 (3/3): the sinusoidal case' --- In this first and last part of our articles about Hopf tori, we will take a sinusoidal curve on $S^2$.  {.r} hopfinverse <- function(q, t){ 1/sqrt(2*(1+q[3])) * c(q[1]*cos(t)+q[2]*sin(t), sin(t)*(1+q[3]), cos(t)*(1+q[3]), q[1]*sin(t)-q[2]*cos(t)) } stereog <- function(x){ c(x[1], x[2], x[3])/(1-x[4]) }  In order to draw a sinusoidal curve on the sphere, I used this equation, that I found on [mathcurve.com](https://www.mathcurve.com/courbes3d/sinusoidespherique/sinusoidespherique.shtml): $$$$x = \frac{\cos u}{\sqrt{1+k^2\cos^2(nu)}} \\ y = \frac{\sin u}{\sqrt{1+k^2\cos^2(nu)}} \\ z = \frac{k \cos(nu)}{\sqrt{1+k^2\cos^2(nu)}}$$$$  {.r} plotSphereWithSinusCurve <- function(){ plotSphereEquator() view3d(0,90) u_ <- seq(-pi, pi, len=1000) k <- 3 for(i in 1:length(theta_)){ u <- u_[i] den <- sqrt(1+k^2*cos(3*u)^2) x <- cos(u)/den y <- sin(u)/den z <- k*cos(3*u)/den points3d(x,y,z, color="blue") } }  ![](figures/SphereWithSinusCurve.png)  {.r} open3d(windowRect=c(50,50,500,500)) view3d(0,90) t_ <- seq(0, 2*pi, len=200) k <- 3 u <- seq(0, 2*pi , len=300) for(i in 1:length(u_)){ u <- u_[i] den <- sqrt(1+k^2*cos(3*u)^2) x <- cos(u)/den y <- sin(u)/den z <- k*cos(3*u)/den circle4d <- sapply(t_, function(t){ hopfinverse(c(x,y,z),t) }) circle3d <- t(apply(circle4d, 2, stereog)) shade3d(cylinder3d(circle3d, radius=0.1), color="purple") }  This time, we obtain a Hopf torus with three lobes (because we took $n=3$ in the formula above).

Below is an interactive rendering with three.js. Go to [this post](https://laustep.github.io/stlahblog/posts/threejsTorus.html) if you want to know how I've drawn the circles.