# Distance between two points on a sphere.

Say there is a sphere on which there is an ant and the ant wants to go to another point. The ant can't definitely travel through the sphere. So it has to travel along a curve. My question is what is the least distance between the two points i.e. distance between 2 points on a sphere.

• These curves are called geodesics, and found using the Christoffel equations.Those are part of great circles, and you can compute the length using integrals. May 29, 2015 at 16:07
• Can you tell me where to find these equations. May 29, 2015 at 16:10
• Or can you tell me where I can learn about them. May 29, 2015 at 16:12
• Learning them alone is, in my opinion, pretty hard if you have not taken any differential geometry class. There are plenty of books. The one I used is Differential Geometry for Curves and Surfaces by Manfredo Do Carmo. May 29, 2015 at 16:14
• Calculating arc lengths of geodesics would be crushing a peanut with a steamroller. :) May 29, 2015 at 16:44

If $a = (a_{1}, a_{2}, a_{3})$ and $b = (b_{1}, b_{2}, b_{3})$ are points on a sphere of radius $r > 0$ centered at the origin of Euclidean $3$-space, the distance from $a$ to $b$ along the surface of the sphere is $$d(a, b) = r \arccos\left(\frac{a \cdot b}{r^{2}}\right) = r \arccos\left(\frac{a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}}{r^{2}}\right).$$ To see this, consider the plane through $a$, $b$, and the origin. If $\theta$ is the angle between the vectors $a$ and $b$, then $a \cdot b = r^{2} \cos\theta$, and the short arc joining $a$ and $b$ has length $r\theta$.

After practicing with Sine/Cosine Rules of spherical Trig and getting their flavor I find ready-made

useful, especially to check special cases.

EDIT1:

The Clairauts Law of geodesics says

$r \sin \beta = a \sin \lambda$

where r is radius, $a$ sphere radius, $\beta$ is path's angle to meridian, and $\lambda$ is co-latitude.