Determining 8 Clearance Points On A Sphere.

 

This problem is that of locating a single point within a quadrant of a hemisphere that is equidistant from the three vertices of the quadrant.

 

This problem is solved by realizing that the 8 clearance points are points at the centroid of the face surfaces of the octahedron inscribed in the sphere which are then projected onto the surface of the sphere along a radial line.  An octahedron is a solid figure that has 8 equilateral triangle faces, 6 vertices, and 12 edges.   See Figure 1.

 

The simplest procedure involves the use of two compasses.

 

Locate the 6 main points as above.

 

Set one compass  to a radius equal to:

 

   ,  where R is the radius of the sphere

 

Set a second compass to the midpoint distance between vertices of the quadrant.

This measurement is given by:

 

 

 ,  where R is the radius of the sphere

 

For a sphere of 50 mm diameter, these measurements are :

r = 35mm, and r’ = 19mm.

For a sphere of 62 mm diameter, these measurements are 44 mm and  24 mm respectively.

 

Place the point of the second compass at each of the three vertices and mark the midpoint of each arc in each of the eight quadrants.

 

Place the point of the first compass on these midpoint marks and draw secondary arcs from each quadrant vertex to the midpoint of the opposite quadrant arc.  Within each quadrant, these secondary arcs will intersect in a point that is equidistant from each of the three vertices of the quadrant.