Girth line equation of Breather

Question

How to find equation of the central shape at $(u=0)$ for the Breather in central plane normal to its axis of polar symmetry?

The Breather is not a surface of revolution.

It is required to find central angle subtended for a segment or the equation whose roots supply ridges /cusps / vertices in the equator/central plane for given $b$. Trying to plot/visualize some regular spiked curved polygons.

Setting $u=0$ results generally in self-intersecting central profiles not periodic in $v,$ except so far found $(b,n)$ combinations (by trial & error) pictured:

Please help in simplifications with a view to find how $b$ or $w$ determines formation of central sections with integral number $n$ of cusps/spikes.

Answer 1

Just set $u=0$ and eliminate the range of u variable:

r := 1 - b^2;
w := Sqrt[r];
u := 0;
denom := b*((w*Cosh[b*u])^2 + (b*Sin[w*v])^2);
breather = {-u + (2*r*Cosh[b*u]*Sinh[b*u])/
     denom, (2*w*Cosh[b*u]*(-(w*Cos[v]*Cos[w*v]) - Sin[v]*Sin[w*v]))/
    denom, (2*w*Cosh[b*u]*(-(w*Sin[v]*Cos[w*v]) + Cos[v]*Sin[w*v]))/
    denom};
ParametricPlot3D[Evaluate[breather /. b -> 0.4],
 {v, -37.4, 37.4},
 PlotStyle -> Thickness[0.02]]

Answer 2

You seem to have accidentally edited-out the link to the definition of the Breather surface (via xahlee.info). For completeness and clarity, here's a formatted version of the parameterization (with parameters $u$ and $v$): $$\begin{align} x &= -u + \frac{r\sinh 2bu}{\delta} \\ y &= \phantom{-u+\;}\frac{2 w \cosh bu (-w \cos v \cos wv - \sin v\sin wv)}{\delta} \\ z &= \phantom{-u+\;}\frac{2w\cosh bu (- w\sin v \cos wv + \cos v \sin wv)}{\delta} \end{align} \tag{1}$$ where $b$ varies from $0$ to $1$, $r := 1-b^2$, $w := \sqrt{r}$, and $\delta := b \left( w^2 \cosh^2b u + b^2\sin^2w v \right)$. I don't know why the author didn't define, say, $b := \cos\beta$, $r := \sin^2\beta$, $w := \sin\beta$; I'll pepper these in when convenient.

When $u=0$, we have $$\delta = \cos\beta \left( \sin^2\beta + \cos^2\beta\sin^2wv \right) = \cos\beta \left( 1 - \cos^2\beta\cos^2wv \right) \tag{2}$$ and $$(x,y,z) = \frac{2\sin\beta}{\delta}\left(0,-w \cos v \cos wv - \sin v\sin wv, - w\sin v \cos wv + \cos v \sin wv \right)\tag{3}$$

In the $yz$-plane, we therefore find $$y^2 + z^2 = \frac{4\sin^2\beta (w^2\cos^2 wv + \sin^2 wv)}{\cos^2\beta(1-\cos^2\beta\cos^2wv)^2} = \frac{4\tan^2\beta}{1-\cos^2\beta\cos^2(v\sin\beta)} \tag{4}$$

If the girth curve is periodically scalloped, we would expect $(5)$ to periodically reach a minimum. Clearly, these minima occur when $$\cos(v\sin\beta) = 0 \quad\to\quad v = \frac{k\pi}{2\sin\beta} \;, k\in\mathbb{Z} \tag{5}$$ The corresponding "polar angles" $\theta_k$ for these minima are given by $$\begin{align} \tan\theta_k = \frac{y}{z} &= \frac{ w \cos v \cos wv + \sin v\sin wv}{ w\sin v \cos wv - \cos v \sin wv} \\[6pt] &= \frac{ w \cos v \cos\frac{k\pi}{2} + \sin v\sin\frac{k\pi}{2}}{ w \sin v \cos\frac{k\pi}{2} - \cos v \sin\frac{k\pi}{2}} \\[6pt] &= \begin{cases} \phantom{-}\cot v, & k \;\text{even} \\ -\tan v, & k \;\text{odd} \end{cases} \end{align} \tag{6}$$ So that $$\theta_k = \begin{cases} \dfrac\pi2-\dfrac{k\pi}{2\sin\beta}, & k \;\text{even} \\[4pt] \phantom{\dfrac\pi2}-\dfrac{k\pi}{2\sin\beta}, & k \;\text{odd} \end{cases} \quad\bmod \pi \tag{7}$$

Since we're reducing modulo $\pi$, it isn't necessarily the case that the $\theta_k$ for consecutive $k$ correspond to consecutive scallops. It's clear that the periodic nature of the girth curve requires $\pi/2/\sin\beta$ to be a rational multiple of $\pi$; that is, we require $\sin\beta$ to be rational, and the evident period of the girth curve should be dictated by (a multiple of) the numerator of $\sin\beta$. (I'm being a little loose here because my brain is tired.)

In OP's second and third examples, we have $$\begin{align} b &= \cos\beta = \frac35 \quad\to\quad \sin\beta = \frac45 \qquad 8\text{-fold symmetry} \tag{8}\\[4pt] b &= \cos\beta = \frac45 \quad\to\quad \sin\beta = \frac35 \qquad 3\text{-fold symmetry} \tag{9} \end{align}$$ For the first example, things are not quite so exact: $$b = \cos\beta = 0.4 \quad\to\quad \sin\beta = 0.9165\ldots \approx\frac{11}{12} \qquad \text{not truly}\; 22\text{-fold symmetry} \tag{10}$$ I'm not sure of the proper way to reconcile the fourth example. I'll leave further consideration to the reader.