Geodesics on torus

Question

Describe the geodesics on Torus

$$\sigma (u,v)= ((a+b \cos u)\cos v, (a+b\cos u)\sin v, b\sin u)$$

First fundamental form for torus is $$b^2 du^2 +(a+b \cos u)^2dv^2$$

Consider unit-speed geodesic $$b^2 \dot u^2 +(a+b \cos u)^2\dot v^2=1$$

By Clairaut's theorem, $\rho \sin \phi =\Omega $ where $\Omega$ is constant, $\phi$ is angle the spiral filament makes to the meridian.

$$\Rightarrow \gamma \text{ is a geodesic. Results are shown below:} $$.

But I do not understand how to choose

$$ 0< \Omega < a-b ~~ \text{or} ~~ \Omega = a-b $$

How to find range of $\rho$ here?

I cannot understand the range shown from minimum radius $\Omega $. Please explain clearly. Thank you :)

Answer 1

By Clairaut's geodesic Law or theorem since

$$ \rho \sin \phi= \Omega,\quad\rho \text{ is always}~>\Omega, ~ \text{ else } \phi \text{ will become imaginary. } $$

Clairaut constant is the red Radius $ \rho= \Omega $ shown.

Filaments or geodesics run populating the yellow regions so that always $ \rho > \Omega$ where torus exists.

Image error $r$... it should actually be $\rho.$

Numerical integration/calculations of odes

$$ \rho \sin \phi = \Omega,~ \frac{d\psi}{ds}=\frac{\cos \phi}{b},(a,b)=(4,1)$$

output the following images.

As $\Omega $ reduces from outer equator $a+b $ down to $a-b $ and further down to zero, we see segments above this lower limit getting populated with criss-crossing geodesic filaments. Case $ \Omega=0$ is for meridional filaments and small values impart a small degree of precession around axis of symmetry.

The results obtained in this calculation agree with the figures in the book.

However for the right figure case of text-book geodesics behavior infinitesimally below $\rho^{-}=\Omega = a-b $ given in the text reference that appears to me as incorrect.

The geodesic goes through the parallel of inner equator tangentially and enters area of the other half torus and back again up to the outer equator as shown in my fig 4 providing some excursion detail of individual space curves..

Torus population with filaments is full on torus whether slightly above or below the $ \rho=\Omega=a-b $ boundary limit. It seems reasonable to assume that at this exact value also we should have the entire torus area covered.

There is no asymptotic behavior noted in my calculation for this case. It seems the book states that the asymptotic limit line along the inner equator is a sort of trap of no return of geodesics but the images below look like it does return from out of a dense bundle of fibers there. Not sure if higher precision in Mathematica calculation is needed. However the plot confirms geodesic goes through extreme positions $ z= \pm ~b $ just as expected.

Understandably this might have been a source of your confusion during self-learning.