Eigenvalue quadric surface method for determining when two ellipsoids share common volume for use in spatial collision detection and avoidance

A method for determining spatial coincidence between a primary object and a secondary object, the method comprising the steps of,

• receiving a primary object 3×3 covariance matrix,
• receiving a secondary object 3×3 covariance matrix,
• receiving a primary 3×1 position vector,
• receiving an object 3×1 position vector,
• extending the primary object covariance matrix into an extended primary object 4×4 covariance matrix with a one in the 4-4 position, zero are inserted in the fourth row and column with a minus one in the fourth-fourth position,
• inverting the secondary object 3×3 covariance matrix into an inverted secondary object 3×3 covariance matrix,
• subtracting the primary object 3×1 position vector from the secondary 3×1 position vector for providing a 3×1 relative position vector,
• extending the inverted secondary object 3×3 covariance matrix by the relative 3×1 position vector into an extended secondary object 4×4 covariance matrix where zero are inserted in the fourth row and column with a minus one in the fourth-fourth position,
• multiplying the extended secondary object 4×4 covariance matrix the inverted covariance matrix by the extended primary object 4×4 covariance matrix for providing a product 4×4 covariance matrix,
• computing four eigenvalues of the product 4×4 covariance matrix,
• examining the four eigenvalues for determining when the four eigenvalues indicate spatial coincidence of the primary object with the secondary object, and
• acting upon the determination of the determining step.