Least–Squares Approximation by Pythagorean Hodograph Spline Curves Via an Evolution Process
The problem of approximating a given set of data points by splines composed of Pythagorean Hodograph (PH) curves is addressed. In order to solve this highly non-linear problem, we formulate an evolution process within the family of PH spline curves. This process generates a one–parameter family of curves which depends on a time–like parameter t. The best approximant is shown to be a stationary point of this evolution. The evolution process – which is shown to be related to the Gauss–Newton method – is described by a differential equation, which is solved by Euler’s method.
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- [AJ05]Aigner, M., Jüttler, B.: Hybrid curve fitting. FSP Industrial Geometry, Report no. 2 (2005), available at: http://www.ig.jku.at
- [FST98]Farouki, R.T., Saitou, K., Tsai, Y.-F.: Least-squares tool path approximation with Pythagorean-hodograph curves for high-speed CNC machining. In: The mathematics of surfaces, VIII (Birmingham, 1998), Info. Geom., Winchester, pp. 245–264 (1998)Google Scholar
- [HKL+99]Hoff, K.E., Keyser, J., Lin, M., Manocha, D., Culver, T.: Fast computation of generalized Voronoi diagrams using graphics hardware. In: SIGGRAPH 1999, New York, pp. 277–286. ACM Press/Addison-Wesley (1999)Google Scholar
- [PLH02]Pottmann, H., Leopoldseder, S., Hofer, M.: Approximation with active B-spline curves and surfaces. In: Proc. Pacific Graphics, pp. 8–25. IEEE Press, Los Alamitos (2002)Google Scholar
- [WPL06]Wang, W., Pottmann, H., Liu, Y.: Fitting B-spline curves to point clouds by squared distance minimization. ACM Transactions on Graphics 25(2) (2006)Google Scholar