Determining 8 Equidistant
Points On A Sphere.
This problem is equivalent to determining the vertices of the inscribed
cube, a solid figure with 6 face surfaces and 8 vertices. See Figure 1.
Locate
the two “top” and “bottom” points as above.
The
length of a side of the cube is:
where R is the radius of the sphere
The
radius of a sphere inscribed in the cube is:
The
distance from the top of the given sphere to a vertex on the cube is:
OR
For a 50mm diameter sphere, the above are a = 29mm, r =
14.5mm and l = 23mm
For a 62mm diameter sphere, the above are a = 36mm, r =
18mm and l = 28mm.
Set a compass to l above.
Place
the point of the compass on the “top” point of the sphere and draw a circle
with the above radius l.
Place
the point of the compass on the “bottom” point of the sphere and draw a circle
with the above radius l.
The
intersections of these circles with the great circles which mark the top and
bottom points will mark the 8 equidistant points, the vertices of the inscribed
cube.