---
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
---

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[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])
}
```

The sphere with the bent equator will be plotted thanks to the following functions.

```{r, echo=FALSE, eval=FALSE}
plotSphereEquator <- function(){
  clear3d()
  view3d(0,90)
  spheres3d(0, 0, 0, radius=1, color="green", alpha=0.5)
  phi <- 0
  theta_ <- seq(0, 2*pi, len=300)
  for(theta in theta_){
    points3d(cos(theta)*cos(phi), sin(theta)*cos(phi), sin(phi), color="black")
  }
}
Rx <- function(alpha) {
  rbind(c(1, 0,          0),
        c(0, cos(alpha), -sin(alpha)),
        c(0, sin(alpha), cos(alpha)))
}
plotSphereSlopedEquator <- function(alpha){
  plotSphereEquator()
  theta_ <- seq(0, 2*pi, len=200)
  phi <- 0
  for(i in seq_along(theta_)){
    theta <- theta_[i]
    rotated <- c(Rx(alpha) %*% c(cos(theta)*cos(phi), sin(theta)*cos(phi), sin(phi)))
    points3d(rotated[1], rotated[2], rotated[3], col="blue")
  }
}
plotSphereSlopedEquator(-pi/8)
snapshot3d("sphereWithSlopedEquator.png")
```

![](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, eval=FALSE}
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:

<!-- ![](figures/hopftorus2.gif) -->

<!-- ![](figures/hopftorus2_anim.gif) -->


<div style="text-align:center">
<img src="./figures/hopftorus2.gif" style="float: left; width: 45%; margin-right: 1%; margin-bottom: 0.5em; border:3px solid pink">
<img src="figures/hopftorus2_anim.gif" style="float: left; width: 45%; margin-right: 1%; margin-bottom: 0.5em; border:3px solid pink">
<p style="clear: both;">
</div>