Is there an algorithm/procedure to place points at equal distances from each other on a sphere?

Answer 1

The shortest distance between points on a sphere occur along the great axis route. That is the section of the sphere that contains the two points as well as the center point. If you want to place a series of points equidistant from each other, you'd first have to define the constant distance, m, then insure that each subsequent point placed is that same distance (m) and a cross section of the sphere at that point also contains the center.
Thats the easiest method. Spherical trigonometry is another approach.

Answer 2

if its physically in front of you, try using string, measuring out the length of it, and wrapping it around and marking off where you land.

Answer 3

Use a protactor and a compas.