| With Loe's idea of singular blending, this paper addresses Non-Uniform Algebraic Trigonometric (NUAT) B-spline curve and surface interpolation methods. The main achievements are as follows:1. Based on the idea to blend a NUAT B-spline curve with a singularly parameterized polygon according to parameterα, a local interpolation method is presented for NUAT B-spline curve, by which a C~2 NUAT B-spline interpolating curve of order 4 without solving a global linear system of equations is generated.2. By using multiple knot insertion techniques, cusps and straight-line segments allow to arise in the NUAT B-spline interpolating curve, and its slackness and smoothness can be controlled by the change of the parameterα. In addition, a G 1 interpolating curve with the local blending parameters is constructed to satisfy the different requirements in geometric modeling flexibly. The NUAT B-spline interpolating curve will reproduce the uniformα-B-spline interpolating one in case of uniform knots.3. Again applying the idea of singular blending to case of surfaces, a NUAT B-spline interpolating blending surface is constructed by a explicit means and maximum distance between the interpolating surface and singular bilinear patch can be adjusted by the blending parameterαsimilar to the case of the curve above. Furthermore, some singular elements such as edges,ruled surfaces and plane can be conveniently embedded in the interpolating blending surface.4. The local interpolation methodd above are extended from order 4 to the case of arbitrary order, the existence for singular piecewise function of arbitrary order is proved, and a unite interpolation formula for odd and even-order is given, which develop and perfect the NUAT B-spline curve and surface interpolation methods.By compared with the most traditional form of surface modeling, the methods presented in this paper are more convenient, more flexible and more effective ,and cover the case of the uniformα-B- spline interpolation one as well. |
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