A trust region algorithm for parametric curve and surface fitting

Use of lp norms in fitting curves and surfaces to data

“…Although the problems are identical, the direct treatment of the problems in the forms given here lead to different algorithms. For the Gauss-Newton method applied to (1), see for example [7,11,12,15], and applied to (2), see for example [2,3,4]. For a unified treatment of these methods, and some comparisons, see [1].…”

Uncertainty associated with form assessment in coordinate metrology

“…For example, a circle with centre coordinates (x 0 , y 0 ) T and radius r 0 is specified by u → C(u, b) = (x 0 + r 0 cos u, y 0 + r 0 sin u) T , where b = (x 0 , y 0 , r 0 ) T . Given data points X, the best fit parameters of the surface is found by minimising some aggregate measure involving the distances d(x i , b) where d(x, b) is the (signed) orthogonal distance from the point x to the surface S(u, b) [1][2][3][4][5][6][7][8][9][10][11][12][13]. For example, Gaussian or Chebyshev criteria are often used:…”

Mathematics for modern precision engineering

“…For standard geometric elements, the distance function d(x,b) can be defined as an explicit function of the parameters but for freeform surfaces the optimal footpoint parameters u * have to be determined using numerical techniques [42][43][44].…”