On a New Distance Approximation for an Implicit Polynomial Manifold Fitting

Abstract

The application of a new approximate point-to-algebraic manifold distance formula is suggested to the geometric approach to curve fitting and surface reconstruction using implicit polynomial manifolds. A brief overview of the fitting methods features for implicit algebraic manifolds is given. To illustrate the possibilities of a new approximate point-to-manifold distance formula, the equidistant curves of the exact distance, Samson’s distance and the present formula are given. A four-step algorithm for implicit algebraic manifold fitting is proposed, using one of the algebraic fitting methods at the initial step, the present approximate formula for the distance finding to calculate the geometric criterion of approximation quality and an optimization method for updating the value of the vector of coefficients of the manifold. The first results of the proposed algorithm on test data are briefly characterized. In conclusion, the tasks and directions for further research are described.