Hi Daniel,
From your picture, it seems that your points are all centered around the [0 0 0] coordinate (and even if they were a little off, I'm sure it would be possible to shift them to ensure this).
So if you were to convert your coordinates to spherical coordinates:
[TH,PHI,R] = cart2sph(x,y,z)
The final output, R, would give the radius of the ball at that point. A perfect sphere would have all points with the same radius. An imperfect (or not properly centered) sphere will have variation.
Your formula for the sphere that covers all the points will simply be determined by max( R ) as:
bigVol = (4/3) * pi * max(R)^3
Perhaps one way to approach the question of deviation from that perfect sphere is to:
- Get an estimation of "percentage of the sphere's surface" that is represented by each coordinate you have. This would be done by some form of Delaunay Triangulation of the points (probably best to use the TH, PHI spherical coords)
- Calculate the volume of that tiny chunk of the sphere via its R radius
- Sum up all the tiny chunks of volumes and compare them to bigVol (keep in mind that it seems you're working with a hemisphere rather than a sphere?)
