【作者】 苏本跃；

【导师】 黄有度；

【作者基本信息】 合肥工业大学 ， 计算数学， 2004， 硕士

【摘要】 本文主要对参数曲线曲面造型的一种新方法——三角多项式曲线曲面进行了深入研究，其内容主要包括T-Bézier曲线曲面、T-B样条曲线曲面、TC-Bézier曲线曲面和TC-B样条曲线曲面。文章最后在双曲函数空间中讨论了HC-Bézier曲线曲面。 本文首先回顾了曲线曲面造型方法的分类以及各自的特点。阐述了CAGD中参数曲线曲面造型的发展历史并介绍了Bzier方法、B样条方法以及非多项式曲线曲面造型方法，后者包括L-样条、螺旋样条、张力样条以及C-曲线等。 文章以Bézier曲线和B样条曲线的特点为基础，在三角函数空间中构造一组具有上述两类曲线特性的三角函数多项式曲线，称其为T-Bézier曲线和T-B样条曲线。它们继承了Bézier曲线和B样条曲线的特点，曲线表示简单、直观。此外由于它们还具有三角函数的优点，故既可以精确表示直线段、二次多项式曲线段又可以精确表示圆弧、椭圆弧等二次曲线以及心脏线、双纽线等超越曲线。特别地，3次均匀T-B样条曲线曲面比同阶均匀B样条(C-B样条)曲线曲面具有更高的光滑度。3次T-Bézier曲线在光滑拼接时也可以达到更高的连续性。最后由于这两类曲线仅由三角函数构成，所以它们较易转化为有理多项式曲线。从而融入到现有的几何造型系统中。 在C-曲线的启示下，本文进一步在T-Bézier曲线及T-B样条曲线中引入控制参数α用以调整曲线形状，构造了另一类自由参数曲线，称其为TC-Bézier曲线及TC-B样条曲线。这两类曲线一方面具有T-Bézier曲线及T-B样条曲线的类似性质和相关二次曲线的精确表示，另一方面由于参数α的引入使得曲线具有更强的表现能力。 文章最后运用同样的方法在双曲函数空间中构造了HC-Bézier曲线。该曲线与TC-Bézier曲线的性质完全类似，此外它既能精确表示直线段又能精确表示双曲线。HC-Bézier曲线中同样具有控制参数α，从而调整曲线形状更加灵活。 对于每一类曲线作者均将它们直接推广到张量积曲面，这些曲面可以精确表示球面、椭球面、双曲面等二次曲面。更多还原

【Abstract】 This paper summaries the researches on the new schemes of parameter curves and surfaces modeling-curves and surfaces modeling of trigonometric polynomial, which includes curves and surfaces of T-Bezier, T-B-spline, TC-Bezier and TC-B-spline. HC-Bé zier curves and surfaces are also discussed in the space of hyperbolic functions in the end.After briefly reviewing the classification and respective characters of curves and surfaces modeling, the paper expatiates its history in CAGD. And then we introduce Bezier, B-spline and non-polynomial curves and surfaces modeling, which include L-splines, helix splines, splines in tension and C-curves etc.By analyzing the characters of Bezier curves and B-spline curves, we construct trigonometric polynomial curves in the space of trigonometric functions, which assume the characters of Bézier curves and B-spline curves. Their representatives are simple and direct. We call them as T-Bezier curves and T-B-spline curves. They not only inherit the advantages of Bezier curves and B-spline curves, also can be used to represent straight lines precisely and some remarkable transcendental curves precisely, such as circular arc, ellipse, cardioids and twisted pair line etc. Especially, the uniform T-B-spline curve of three degrees is smoother than B-spline curve and C-B-spline curve of the same order. When connected smoothly, T-Bezier curve of three degrees can attain more superior continuity. At last, the T-Bézier curves and T-B-spline curves can be converted to rational curves so easily that they can be merged into current geometry modeling rapidly.Furthermore, with the illuminating of C-curves, the paper construct another freeform parameter curves and surfaces by introducing parameter a in control. We call them as TC-Bezier curves and TC-B-spline curves. On the one hand, they possess the similar characters of T-Bezier curves and T-B-spline curves. On the other hand, by introducing the parameter a, they act the wonderful ability of representation.At last, we construct hyperbolic polynomial curves in the space of hyperbolic functions. We call them as HC-Bezier curves. Similarly, they not only can represent straight lines precisely, but also can represent quadric curve precisely, such as hyperbola and so on. Parameter a in control is also used in this kind of curve.The generation of tensor product surfaces of every kind of curves is straightforward; these corresponding tensor product surfaces also contain many special surfaces, including spherical surface, ellipsoid, hyperboloid etc.更多还原