I have the following curve specified in spherical coordinates $(r, \theta, \phi)$*:
$$K^2 \tan^2(\phi-\phi_0) = \tan^2 \theta – K^2 \sec^2 \theta, \quad r=1,$$
where $K$ and $\phi_0$ are constants.
I want to try and show that this specifies a great circle on the unit sphere. I'm not sure how to intuitively see this is the case.
I have tried to convert to Cartesian coordinates to arrive at an equation of the form $f(x,y,z)=0$, but after doing this I still do not see how to arrive at the conclusion that this is a great circle.
*I am using the convention that $\theta \in [0,\pi]$ is the inclination and $\phi \in [0,2π)$ is the azimuth, that is, $x=r \sin \theta \cos \phi$, $y=r \sin \theta \sin \phi$, $z=r \cos \theta$.
Best Answer
A point on the unit sphere has Cartesian coordinates
$P = (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta) $
Starting with the given equation
$K^2 \tan^2(\phi - \phi_0) = \tan^2(\theta) - K^2 \sec^2(\theta) $
Using the identity $\sec^2(\theta) = 1 + \tan^2(\theta) $ the above equation becomes
$K^2 \tan^2(\phi - \phi_0) = (1 - K^2) \tan^2(\theta) - K^2 $
Multiply through by $\cos^2(\phi - \phi_0) \cos^2 \theta $, you get
$ K^2 \sin^2(\phi - \phi_0) \cos^2 \theta = (1 - K^2) \sin^2 \theta \cos^2 (\phi - \phi_0) - K^2 \cos^2(\phi - \phi_0) \cos^2 \theta $
This re-arranges into
$ K^2 \cos^2 \theta ( \sin^2(\phi - \phi_0) + \cos^2(\phi - \phi_0) ) = (1 - K^2) \sin^2 \theta \cos^2(\phi - \phi_0) $
which reduces to
$ K^2 \cos^2 \theta = (1 - K^2) \sin^2 \theta \cos^2(\phi - \phi_0) $
The solution of the above equation is
$ K \cos \theta = \pm \sqrt{1 - K^2} \sin \theta \cos(\phi - \phi_0) $
Here, we have to assume that $| K | \le 1 $
Expanding the right hand side
$ K \cos \theta = \pm \sqrt{1 - K^2} \sin \theta \left( \cos \phi \cos \phi_0 + \sin \phi \sin \phi_0 \right) $
Now recall the expression for a point $P$ on the unit sphere, then the above equation says
$ [ \pm \sqrt{1 - K^2} \cos \phi_0 , \pm \sqrt{1 - K^2} \sin \phi_0 , - K ] \cdot P = 0 $
which is of the form
$ N \cdot P = 0 $
and this is an equation of a plane passing through the origin. Since $P$ is on the unit sphere, then the intersection of the plane and the sphere is a great circle. Taking the plus and minus cases, this implies that the original equation corresponds to two possible great circles.