Research On Curves And Surfaces Modeling Of Trigonometric Polynomial In CAGD

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.