Optimal approximate conversion of spline curves and spline approximation of offset curves (Q1116632): Difference between revisions

Parametric polynomial representation with different polynomial bases and maximum polynomial degrees is used for CAD systems for free-form curves and surfaces modelling. This involves conversion from one polynomial base to another. By direct matrix multiplication, one can achieve such a conversion whenever the number of degrees of polynomial terms in both representations are equal. If two systems do not allow for the same maximum polynomial degrees, then approximate conversions of high order function into low order functions or vice versa is inevitable. This causes an approximation error which has to be minimized. This has been achieved by \textit{L. Dannenberg} and \textit{H. Nowacki} [Comput. Aided Geom. Des. 2, 123-131 (1985; Zbl 0577.65005)] using an error estimate due to \textit{C. de Boor} [A practical guide to splines. (1978; Zbl 0406.41003)] and its applications due to \textit{G. Hölzle} [Comput. Aided Des. 15, 295-296 (1983)]. The first author [ibid. 17, 77-82 (1985)] has proposed a conversion method using geometric continuity of orders 1 and 2 and parametric optimization. In the present paper, this method is extended to geometric continuity of orders 3 and 4. A more effective nonlinear optimization algorithm and a spline splitting algorithm are also introduced.