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.
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$\begingroup$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 $r\theta$.
$\endgroup$ 0 $\begingroup$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.
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