Loxodrome parametric equations

Question

I have been trying to understand HOW one arrives at the equations

$x=cos(t)cos(c)$

$y=sin(t)cos(c)$

$z=−sin(c)$ of the loxodrome.

I can see that if the transformation to spherical coordinates is

$x=sinϕcosθ$

$y=sinϕsinθ$

$z=cosϕ$

then the "loxodromic equations" above are the derivative of the transformation with respect to $ϕ$ and then replacing $\theta$ with $t$ and $\phi$ with $c$.

Could someone make sense out of this? Thanks!

Answer 1

Here is the answer...

The parametric equations

$x=\cos(t) \cos(c)$

$y=\sin(t) \cos(c)$

$z= -\sin(c)$

are actually "oblate spheroidal coordinates" for the special case where the oblate spheroid is actually a sphere (see the definition of "oblate spheroidal coordinates" at Wikipedia, the parameter $\mu$ is assumed positive there, but is negative above; when its modulus is large enough, the hyperbolic sin and cos have equal values, thus acting as a "radius" of the the oblate spheroid) and the inclination is measured with respect to the horizontal plane.