Spherical Distance Formula?

Question

So suppose you have two spherical coordinates $(a,b,c)$ which both have the same $a$ but varying $b,c.$ Is there a formula for the distance having to go around the globe to reach the other side?

So what spherical coordinates are is $(x,y,z)$ represents a sphere with radius $x,$ shifted $y$ radians on the x-y axis (like polar form) and shifted $z$ radians on the $z$ axis from its initial top.

One example I have is $\left(12,\frac{2\pi}3,\frac{\pi}4\right),\left(12,\frac{3\pi}8,\frac{5\pi}6\right)$.

I've been thinking of maybe finding a circle with the greatest radius on the sphere, and finding the distance in this circle.

Answer 1

Not a formula, but a method:

• Convert each point to Cartesian (rectangular) coordinates, and denote the resulting position vectors $\vec{r}_1$ and $\vec{r}_2$.
• Compute the angle $\theta$ between $\vec{r}_1$ and $\vec{r}_2$ using the formula $\cos \theta = \frac{\vec{r}_1\cdot \vec{r}_2}{||\vec{r}_1||\vec{r}_2||}$. (In the case $a=1$ this is particularly simple.)
• Finally, the arc joining your two points on the sphere of radius $a$ will have length equal to $a\theta$.

If you really want to, you can use the above to get a general formula in terms of the spherical coordinates of the points, but I'm not sure there's a lot of value-added to that.