---
author: Stéphane Laurent
date: '2018-08-25'
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, asymptote, geometry, maths'
title: Parametric Hopf torus
---

Identify $S^3$ with the unit quaternions and $S^2$ with the unit
quaternions with a null $i$-part component. Then the Hopf map is
expressed as $$
\begin{cases} \Pi \colon & S^3  \to S^2 \\ 
& q  \mapsto \tilde{q} q \end{cases}
$$ where the quaternion $\tilde{q}$ is obtained from $q$ by negating its
$i$-th component.

``` {.r}
library(onion)
Pi <- function(q){
  qtilde <- q
  i(qtilde) <- -i(q)
  qtilde*q
}
Pi(quaternion(Re=1, i=2, j=3, k=4))
##    [1]
## Re -20
## i    0
## j   22
## k   -4
```

Choose a curve $p \colon [a,b] \to S^2$ and a *lift* of $p$, that is a
map $y \colon [a,b] \to S^3$ such that $p = \Pi \circ y$.

For our illustration, we will take the closed curve
$p \colon [0,2\pi] \to S^2$ defined by $$
\begin{cases}
p_1 \colon t \mapsto \sin\bigl(h\cos(nt)\bigr) \\
p_2 \colon t \mapsto \cos(t)\cos\bigl(h\cos(nt)\bigr) \\
p_3 \colon t \mapsto \sin(t)\cos\bigl(h\cos(nt)\bigr)
\end{cases}
$$ where $n$ is an integer.

``` {.r}
h <- 0.4
n <- 3
p <- function(t) c(sin(h*cos(n*t)),
                    cos(t)*cos(h*cos(n*t)),
                    sin(t)*cos(h*cos(n*t)))
```

The lift is easy to get: simply take $$
y(t) = \frac{\bigl(1+p_1(t),0,p_2(t),p_3(t)\bigr)}{\sqrt{2\bigl(1+p_1(t)\bigr)}}.
$$

Let's check:

``` {.r}
y <- function(t){
  c(1+p(t)[1], 0, p(t)[2], p(t)[3]) / sqrt(2*(1+p(t)[1]))
}
Pi(as.quaternion(y(1), single=TRUE))
##           [1]
## Re -0.3857282
## i   0.0000000
## j   0.4984896
## k   0.7763516
p(1)
## [1] -0.3857282  0.4984896  0.7763516
```

Ok.

The *Hopf cylinder* of $p$ is then $(t,\phi) \mapsto e^{i\phi}y(t)$, and
we say it's a *Hopf torus* when the curve $p$ is closed. This is a
parametric representation of a Hopf torus.

Here, the integer $n$ determines the number of lobes of the Hopf torus,
and\
the parameter $h$ controls the shape.

Now, apply the function $(t,\phi) \mapsto e^{i\phi}y(t)$ and then apply
the\
stereographic projection.

Below is a R code after we got rid of the quaternions.

``` {.r}
h = 0.4
n= 3 # number of lobes
F <- function(t,phi){
  p2 <- cos(t) * cos(h * cos(n*t))
  p3 <- sin(t) * cos(h * cos(n*t))
  p1 <- sin(h * cos(n*t))
  ## alternatively, 
  # den = sqrt(1+h^2*cos(n*t)^2);
  # p2 = cos(t)/den;
  # p3 = sin(t)/den;
  # p1 = h*cos(n*t)/den;
  ##
  yden <- sqrt(2*(1+p1))
  y1 <- (1+p1)/yden
  y2 <- p2/yden
  y3 <- p3/yden
  cosphi <- cos(phi)
  sinphi <- sin(phi)
  x1 <- cosphi*y1
  x2 <- sinphi*y1
  x3 <- cosphi*y2 - sinphi*y3
  x4 <- cosphi*y3 + sinphi*y2  
  return(c(x1/(1-x4), x2/(1-x4), x3/(1-x4)))
}
```

Now we can use the `misc3d` package to plot the projected Hopf torus.

``` {.r}
fx <- Vectorize(function(u,v) F(u,v)[1])
fy <- Vectorize(function(u,v) F(u,v)[2])
fz <- Vectorize(function(u,v) F(u,v)[3])
library(misc3d)
parametric3d(fx, fy, fz, 
             umin = 0, umax= 2*pi, 
             vmin = 0, vmax = 2*pi, 
             n = 300, color = "red")
```

![](figures/HopfTorus_blog.png)

Since I like Asymptote, I also provide the Asymptote code.

``` {.c}
settings.render = 4;
settings.outformat="pdf";

size(500,0);
import graph3;
import palette;

real h = 0.4;
real n = 3;
triple F(pair uv){
  real t = uv.x;
  real phi = uv.y;
  real p2 = cos(t) * cos(h * cos(n*t));
  real p3 = sin(t) * cos(h * cos(n*t));
  real p1 = sin(h * cos(n*t));
  real yden = sqrt(2*(1+p1));
  real y1 = (1+p1)/yden;
  real y2 = p2/yden;
  real y3 = p3/yden;
  real cosphi = cos(phi);
  real sinphi = sin(phi);
  real x1 = cosphi*y1;
  real x2 = sinphi*y1;
  real x3 = cosphi*y2 - sinphi*y3;
  real x4 = cosphi*y3 + sinphi*y2;  
  return (x1/(1-x4), x2/(1-x4), x3/(1-x4));
}

splinetype[] Notaknot={notaknot,notaknot,notaknot};
surface s=surface(F,(0,0),(2pi,2*pi),116,116,Notaknot,Notaknot);
s.colors(palette(s.map(abs),Gradient(8192,yellow,green)));
draw(rotate(-20,(0,1,0))*rotate(-45,(0,0,1))*rotate(90,(1,0,0))*s);
```

![](figures/HopfTorusParametric00_k1_R.png)

And I also like `three.js`:

<iframe src="../frames/threejs_HopfTorusParametric.html" width="100%" height="500px" scrolling="no" frameborder="0">
</iframe>