I am trying to use CGA to solve problems in the context of marker clusters used in optical motion analysis. Four markers (points in 3d) always forms a sphere and can be easy represented with a opns sphere in CGA. For such round objects an euclidean direction can be calculated. What is the meaning of that object and how it depends on the order/arrangement of the four points in 3d. The hope is that I can use the direction to somehow identify the points configuration (by usage together with the radius of the sphere). In the case of a circle the meaning of the direction is clear.
Any help?
The direction of the opns sphere is of grade 4: e1^e2^e3^inf and I want to understand die meaning of its value.
Hi Oliver,
What you are computing is in fact a weighted sphere; this is perhaps most easily seem by its dual, which will be of the form \alpha (c - \rho^2 n_\infty/2), so the factor you are interested is \alpha \rho^2. That shows that the radius is in there, and the factor \alpha can be interpreted as a measure of the numerical reliability of the spanned sphere due to the points. I would imagine that it is proportional to the volume of the tetrahedron spanned by the points: smaller as they are more coplanar.
A similar factor (but 1 dim lower) occurs as the denominator in the expression for the circumcenter of 3 points (see GA4CS, pg 457).
Cheers, Leo