11
$\begingroup$

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.

$\endgroup$
6
  • $\begingroup$ 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. $\endgroup$
    – mich95
    May 29, 2015 at 16:07
  • $\begingroup$ Can you tell me where to find these equations. $\endgroup$ May 29, 2015 at 16:10
  • $\begingroup$ Or can you tell me where I can learn about them. $\endgroup$ May 29, 2015 at 16:12
  • $\begingroup$ 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. $\endgroup$
    – mich95
    May 29, 2015 at 16:14
  • 3
    $\begingroup$ Calculating arc lengths of geodesics would be crushing a peanut with a steamroller. :) $\endgroup$ May 29, 2015 at 16:44

2 Answers 2

9
$\begingroup$

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
2
$\begingroup$

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

formula

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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.