Shortest Distance Between Two Points on a Sphere

Imagine a sphere in three-dimensional space (also known as Euclidean Space), and imagine adding two points on the surface of the sphere (with cartesian coordinates; aka (x,y,z) coordinates for the two points). How would we find the shortest distance along the surface of the sphere between the two points? Like the distance between A and B along the surface of the sphere.

This involves nothing more than trigonometry, Pythagoras' Theorem and the equation for the length of the arc.

So we know from Pythagoras:

And for three dimensions:

To put this in terms of the coordinates of A and B:

remember x is the difference between the x coordinates of the 2 points and the same with y and z

d is the straight-line distance between the 2 points, not the distance along the surface

Now if you were to place a single point on a sphere and another point on the sphere, you could rotate the sphere to align the two points along the same axis, and if you were to follow the line along the sphere to the other point, well that should give you the shortest distance.

For example:

The shortest distance between A and B is the line highlighted in black. This shows us that the arc for the shortest distance is the arc of the circle with the same radius as the sphere; this is due to the radius of the circle shown (A and B) being equal to the radius of the sphere.

And the arc length is given by:

where theta is the angle between the two points from the centre, in the example above it is 90 degrees

Since we know the direct distance between A and B (d), we can form an isosceles triangle with the radii connecting the points to the centre, and the direct distance between the 2 points. Like:

We can find the angle theta from the cosine rule now:

We can now substitute this into the equation for arc length, and we can substitute the earlier formula for the direct distance into the one above, and the resulting formula looks like this:

Now, this is our final formula, for the shortest distance between two points along the surface of the sphere as a function of the (x,y,z) coordinates for both points and the radius of the sphere.

I hope you've enjoyed the post, I would like to credit GeoGebra and the website desmos, as they have proved very helpful over the last posts. Be sure to check them out!-naBla5040