I am working on an extension of 3.2 #19a from Do Carmo that asks what the relationship between the geodesic torsions is at the point of intersection between two curves, given that they intersect perpendicularly. I understand how to prove #19, but am having a hard time with the intuition behind the relationship between geodesic curvature for the different curves.
For reference, Do Carmo #19a proves the result that:
$\tau_g=(k_2-k_1)\cos\phi \sin\phi$
where $\phi$ is the angle from $e_1$ to $t$ and $\tau_g$ is the geodesic torsion.
Intuitively, I want to say that because for two curves $\alpha(t),\beta(t)$ that intersect perpendicularly at $t=0$, $\phi_\alpha=\phi_\beta\pm \pi/2$, and from that $\cos(\phi_\beta) = \mp\sin(\phi_\alpha)$ and $\sin(\phi_\beta) = \cos(\phi_\alpha)$, then $\tau_{g,\alpha} = \pm\tau_{g,\beta}$.
This feels too simple and I feel like I am missing part of the analysis, though I don't know where I am going wrong.
Can anyone offer me any guidance?