二次三角bézier曲线的保形插值分析-conformal interpolation analysis of quadratic triangular b é zier curve.docx

摘要本文主要利用二次三角多项式研究了CAGD中平面曲线的保形插值问题，给出了几种保形插值新算法。主要的研究工作及结果如下：首先，给出了二次三角Bézier曲线的显式表达式，并研究了曲线的性质。其具有与三次Bézier曲线类似的性质：端点性质，几何不变性，对称性，凸包性，变差缩减性及保凸性。相应实例表明，二次三角Bézier曲线比三次Bézier 曲线更靠近控制多边形，具有更好的保形效果。其次，讨论了二次三角Bézier曲线的拼接，并构造了两种均达到C3连续，且使曲线保形插值的算法：利用拼接思想给出了可整体调节曲线形状的算法及分段构造可局部调节曲线形状的算法，分别给出了相应实例。再次，基于二次三角Bézier曲线构造了有理二次三角Bézier曲线，并研究了曲线的性质。此曲线具有与三次有理Bézier曲线类似的性质：端点性质，几何不变性，凸包性，变差缩减性及保凸性。讨论了有理二次三角Bézier曲线的几何连续性拼接和参数连续性拼接，并利用拼接的思想分别给出了曲线达到C2 连续及G3连续的保形插值算法。两种方法均无需求解方程组，计算简便，且曲线的形状都可作局部调节。相应实例表明：G3连续的保形插值算法构造的有理曲线保形效果最好，具有极大的应用价值。最后，对全文内容进行了总结，并提出了有待研究的问题。关键词：保形插值；二次三角Bézier曲线；有理二次三角Bézier曲线；C3连续；G3连续；C2连续ABSTRACTThisthesismakesmainuseofquadratictrigonometricpolynomialtostudythe problemofshapepreservinginterpolationforplanercurvesinCAGDarea,and it providessomealgorithmsoftheshapepreservinginterpolation.Themainworkand resultsareoutlinedasfollows:First,itgivestheexplicitexpressionsofquadratictrigonometricBézier curves,andstudiesthe propertiesofthe curves,whicharesimilartothatofcubic Béziercurves:propertiesofendpoints,geometricalinvariability,symmetry,convex hullproperty,thevariationdiminishingpropertiesandconvexity-preserving.The correspondingexamplesshowthatquadratictrigonometricBéziercurves,whichare closertothecontrolpolygoncomparedwiththecubicBéziercurves,haveabetter shapepreservingeffect.Secondly,itdiscussesthejoiningbetweenadjacentquadratictrigonometric Béziercurves,andgeneratestwoalgorithmstoconstructtheshapepreserving interpolationcurveswhichreachC3continuity:onethattakesadvantageofthe joiningbetweenadjacentcurvesmeetstheglobaladjustmentofthecurves’shape;the otherusingthepiecewisecurvesallowsthelocalmodificationofcurves’shape.Thecorrespondinggraphexamplesaregivenrespectively.Thirdly,itconstructsrationalquadratictrigonometricBéziercurvesbasedonthe quadratictrigonometricBéziercurves,andstudiesthepropertiesofthecurves,which aresimilartothatofrationalcubicBéziercurves:propertiesof endpoints,geometricalinvariability,convexhullproperty,thevariationdiminishing propertiesandc