Details

A roll in the horizontal plane of length about an axis at angle with the axis produces the following change in orientation:

.

We want to choose the pair that moves a point from the starting latitude-longitude pair to the desired latitude-longitude pair.

First, we compute the 3D location of the point from the latitude-longitude pair :

.

Let and be the start and end coordinates.

The angle to roll along is given by , which is undefined if the coordinates overlap. If they overlap and any angle works. Otherwise, set .

The length of the roll is determined by the angle between the start and ending latitude-longitude pair with respect to the intersection of the line in the - plane between the start and end points and a perpendicular line to the origin. If is the squared - distance between the start and end points, this intersection point is given by

.

If , the intersection point is the origin.

If and are the vectors to the starting and ending north poles from this intersection point, then we can use a property of the dot product to calculate the angle , which satisfies .