Given two points on a unit sphere, how to express their angular difference in spherical coordinates?

Question

Given two points on a unit sphere in spherical coordinates representation ($\theta$ is longitude, $\phi$ is latitude in $[0,\pi]$), I want to express their angular difference in spherical coordinates.

In detail: given two points $p_1,p_2$ on the meridian, the first is under the second. Making the first the new north pole but leaving the meridian at $\theta =0°$ should give me a spherical coordinate with $\phi = |\phi_1 - \phi_2| $ which seems obvious, but the new longitude $\theta$ needs to become $180°$ although $\theta_1 = \theta_2 = 0$ (meridian).

If the answer is to convert both coordinates to Euclidian (X,Y,Z), then subtract and then reconvert to spherical coordinates I would not be surprised. But I would be very glad, if you could state why such a difference as in $ \mathbb{R}^{3}$ is not possible in spherical coordinates. What property do they miss?

Edit: these images should clarify my intentions.

The left image shows an angular difference. But I understand that a coordinate-wise difference is not sufficient. To show an example, please look at the right image.

$A$ becomes the new north pole. Thereby $B's$ polar coordinate changes to $\theta(B-A)$. But additionally the azimuthal angle of $B'$ becomes $\phi(B)=$ 180°. Imagine another extreme case where $\phi(B)$ started with 90°. It would end up at the same 90° after making $A$ the new north pole.

Answer 1

Given two points on a sphere $(x_1,y_1,z_1)=(\sin \phi_1 \cos \theta_1,\sin \phi_1 \sin \theta_1,\cos \phi_1 )$ and $(x_2,y_2,z_2)=(\sin \phi_2 \cos \theta_2,\sin \phi_2 \sin \theta_2,\cos \phi_2 )$. Take the dot product of these to obtain the cosine of the angle $\alpha$ between them relative to the center of the sphere. We have \begin{eqnarray*} \cos \alpha = \cos \phi_1 \cos \phi_2 +\sin \phi_1 \sin \phi_2 ( \cos \theta_1 \cos \theta_2 +\sin \theta_1 \sin \theta_2) \\ \cos \alpha = \cos \phi_1 \cos \phi_2 +\sin \phi_1 \sin \phi_2 \cos (\theta_1 - \theta_2 ). \end{eqnarray*} \begin{eqnarray*} \alpha = \color{blue}{\cos^{-1} \left( \cos \phi_1 \cos \phi_2 +\sin \phi_1 \sin \phi_2 \cos (\theta_1 - \theta_2 )\right)}. \end{eqnarray*}

Answer 2

The case you described is a change from one spherical coordinates system to another by exchanging the former zenith axis with a new direction expressed as a point on the sphere.

You might want to have a look at the solution of these questions:

• Converting between spherical coordinate systems
• Transforming from one spherical coordinate system to another

Another method that worked for me is to do two rotations:

1. Rotate point $B$ on the sphere around the zenith axis for $\phi(\,A\,)$ degrees. This positions $A$ on the meridian and makes $\phi(\,B\,)=\phi(\,B-A\,)$. This is a normal difference operation.
2. Rotate $B$ on the sphere along the meridian around the $(\theta=90°,\phi=90°)$ axis for $\theta(\,A\,)$ degrees. This rotates $A$ along the meridian and puts it in place of the north pole. The relation between $A$ and $B$ will be preserved and your example cases are all covered. I cannot offer a formula for that, maybe others can.

If you need to implement the rotation of a point around an arbitrary axis, it is necessary to use quaternions which should be provided by your language's libraries. For C/C++ I recommend glm. I computed the rotation in XYZ-coordinates and transformed the resulting unit vector back to spherical coordinates.