the mathworld web site http://mathworld.wolfram.com/SphericalSpiral.html claims that the loxodrome is given by the parametric equations

##x=cos(t) cos(c)##

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

##z=-sin(c)##

Why so?

Now, as far as I can see, since the spherical coordinates are

##x=sin\phi cos\theta##

##y=sin\phi sin\theta##

##z=cos\phi##

Then the loxodrome equations look like the derivative of the radius vector ##{\mathbf r}## with respect to the zenith angle ##\phi## in spherical coordinates, namely

##\frac{dx}{d\phi} = cos \phi cos \theta##

##\frac{dy}{d\phi} = cos \phi sin \theta##

##\frac{dz}{d\phi} = -sin \phi##

where ##\theta, \phi## are the usual spherical angles, but in the loxodrome equations they are just replaced with ##t## and ##c## respectively.

Would anyone know why is the loxodrome defined in exactly the way shown above?

Thanks!