Show that loxodrome helices on a cone of revolution project on a plane perpendicular
to their axes (the base) as logarithmic spirals and then show that the intrinsic
equations of these conical helices are:
$$k=\frac{1}{as} \ \ \tau=\frac{1}{bs}$$
I'm trying to solve this problem but I lack a lot of knowledge about how to approach this. This is the first time I have heard about helix on a cone so I don't know anything about it nor do I know how to derive its intrinsic equations.
What I do know is that the projection of a helix on a plane perpendicular to its axis has parallel normal vector to the corresponding normal vector of the helix and it's curvature is $k_1= \frac{k}{\sin^2 {\alpha}}$ where $\alpha$ is constant angle that the tangent vectors make with a fixed line (the generators of the cone in this case I believe). The problem is that I have no idea what the curvature of the original helix is to relate it to its projection.
I could either try to use the relations between the curvatures or another way is to prove that the position vectors of the projection makes constant angle with its tangent vector, the projection then is logarithmic spiral.
The following is also provided as a hint in my book which was of very little help since I don't even know what $R,R_1,s,s_1$ in this case are:
These helices also intersect the generating lines of the cone at constant
angles (loxodromes); their projections on the base intersect all radii at constant
angles; from the theorem on the projection of a helix on the plane
follows that both R and s are proportional to $R_1 \ s_1$ and $R_1$ is proportional to
$S_1$
Best Answer
Adopting $ (r,\theta,z) $ cylindrical coordinates $$\phi= \alpha=const, \psi= \gamma= const$$
where $\alpha$ is semi-vertical cone angle and $\psi$ is what you indicated as $\alpha$.
Using relations among differential surface element lengths $ (dr, r d\theta, dz)$
$$ \frac{dr}{d\theta}=\frac{dr}{ds} \cdot\frac{ds}{d \theta}= \sin \alpha \cos \psi \cdot\frac{r}{ \sin \psi} = r \cot \gamma \sin \alpha$$
i.e., $$ \frac{dr}{r d\theta}=\sin \alpha \cot \gamma \cdot \tag 1 $$
which integrates to
$$ r/r_{min}= e^{\sin \alpha \cot \gamma\,\cdot \theta} \tag 2 $$
with $r=r_{min}$ boundary conditions at $\theta=0,$
which is a log spiral.
and next
$$ z=r \cot \alpha$$
from which you can calculate $(\kappa,\tau)$ on arc length basis, you can take it further parametrically.