Stitching B - Spline Curves Symbolically Stitching B - spline

Stitching or merging B-spline curves is a frequently used technique in geometric modeling, and is usually implemented in CAD-systems. These algorithms are basically numerical interpolations using the least squares method. The problem, how to replace two or more curves which are generated separately and defined as B-spline curves, has well functioning numerical solutions, therefore, relatively few papers have been published about this topic. In [6] and [3] methods for approximate merging of B-spline curves and surfaces are given. In [4] one of the symbolical algorithms is described, which extends B-spline curves by adding more interpolation points one by one at the end of the curve. In [5] the construction of a covering surface is shown for unifying more B-spline surfaces. We approach the stitching problem from a geometrical point of view, and represent a symbolical solution to compute the control points of the new curve from the control points of the two given curve segments and appropriate interpolation conditions. This symbolical solution is stable, it can be used generally for any two given curves. The error of the interpolation depends on the curvatures of the input curves. Larger difference in their curvatures raises the error. In order to reduce the error, two of the new control points are adjusted by fairing conditions using the concrete numerical data. This computation requires minimization of quadratic functions leading to solve linear equations. In this way we avoid non-linear optimization problems. Applying fairing functions for modifying the shape and the properties of curves and surfaces is a standard technique. In [7], [8] and [9] constructions of B-spline surfaces with boundary conditions are presented using fairing functions. Finally, merging of B-spline surface patches are shown applying the developed curve stitching method for their parameter curves.