Nonplanar curve and surface estimation in 3-space

The problem of minimal parameter representation and estimation for complex planar and nonplanar curves, and surfaces is considered. The representation is based on concepts from algebraic geometry: a surface is the set of roots of a polynomial of three variables, and a curve is the intersection of two different surfaces. It is shown that the surfaces of an interesting complex of objects in three-space can be represented by single high degree-polynomials, and a similar statement applies to complex curves in three-space. An approximate expression for the mean-square distance from a set of points to a curve or surface is developed, not only for quadratic surfaces, but also for surfaces and curves defined by polynomials of higher degree. A computationally efficient algorithm is presented to carry out the minimization without using nonlinear optimization techniques