The quadric surface fitting through cloud points was described as the constrained nonlinear optimization. The calculation of initial fitting parameters and optimization algorithm were summarized, and then the influence of point coordinate measurement error on fitting parameters and evaluation of surface contour was researched. The methods to assess their uncertainty through analysis and Monte-Carlo simulation were discussed. It was found that the uncertainty of fitting parameters and surface contour could be evaluated reliably only through Monte-Carlo simulation for measurement data from actual surfaces. The simulation and actual measurement experiments were conducted, which suggested that the methods of quadric surface fitting and uncertainty assessment presented be correct.
Dai M, Newman T S. Hyperbolic and parabolic quadric surface fitting algorithms-comparison between the least squares approach and the parameter optimization approach . 1998-02 http://www.cs.uah.edu/tech-reports/TR-UAH-CS-pdf