Abstract
In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary polynomials. In this article (part 2), the notion of B-spline is explained by means of the transition from a representation of a polynomial in the monomial basis (ordinary polynomial) to the Lagrangian form, and from it to the Bernstein form, which finally yields the B-spline representation. Moreover, the direct relation between the B-spline parameters and the parameters of a polynomial in the monomial basis is derived. The numerical stability of the spline approximation approaches discussed in part 1 and in this paper, as well as the potential of splines in deformation detection, will be investigated on numerical examples in the forthcoming part 3.
Highlights
The use of point clouds derived from areal measurement methods, such as terrestrial laser scanning or photogrammetry, results in the necessity to approximate them by a curve or surface that can be described using a continuous mathematical function, often by means of splines
A starting point for advanced considerations in engineering geodesy are almost always the formulas for B-spline curves and B-spline surfaces given in the textbook by Piegl and Tiller [2], where the functional values of the B-spline basis functions are recursively computed according to the formulas by de Boor [3] and Cox [4]
Point clouds derived from areal measurement methods, such as terrestrial laser scanning or photogrammetry, are often approximated by a continuous mathematical function for further analysis, such as deformation monitoring
Summary
Polynomial to B-spline is shown in Appendix B. With the monomial basis functions φi ( x) = xi , i = 0,1, , n , We obtain for the parabola with n = 2 from Error!
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