Research On Rational Construction For Offset And RMF In CAGD

Constructing rational offset and rational Rotation-Minimizing Frame of Bezier curve or B-spline curve during rendering sweep surface which are compatible with CAD system are two important problems in CAGD. In this paper, We do deep research on rendering offset to planar curve and RMF to space curve depending on PH-spline(or OR-spline) approximation for planar curve and PH-spline approximation for space curve.So far, the algorithms for generating offset are categorized into approximation and exact method separately. The first one is mainly based on employing proper function with parameter as an approximation for the offset to the original curve, which usually causes complicated process of solving linear or nonlinear equation groups, nonlinear computing , integral and output unrobust solution. The unique exact algorithm is involved in identifying offset rationality after analyzing an complicated equation in term of the resultants of two polynomials that are the parametric coordinate components of the generator, thus this method has relatively narrow application scope. In practice, we hope the designed curve be rational parametric offset, but on the contrary, the most current systems are only compatible with polynomials which rare have offset in term of rational polynomial forms. In this paper, we proposed an new type of algorithm for rendering offset based on PH-spline or OR-spline approximaton for the original curve after combine approximation and exact methods. (1) Generating offset based on PH curve. We employ G~1 cubic PH-spline , C~1 quintic PH-spline and blending PH-spline which maybe containing G~1,G~1C~2,G~2 PH-splines approximation for the given orignal Be zier or B-spline curve with the bound error. Further renders exact rational offset to the PH-spline ,which can be regarded as an approximate offset to the Bezier curve. The errors between PH-spline and the Bezier curve , the offset to PH-spline and the offset to the given Bezier curve are also estimated. (2) Generating offset based on OR curve. OR curve is polynomial and rational parametric plane curve whose offsets are rational. We here utilize the necessary and sufficient condition for properly...