基于结式方法的代数曲面拼接

Resultant-based method of algebraic surface blending

以同伦连续映射理论为基础,构造代数曲面拼接应该满足的代数方程组。然后,利用结式方法消去相关变元得到拼接曲面方程。两代数曲面拼接时,方程组是两个关于单位区间变元的方程。利用Sylvester结式消去该变元即可得到曲面拼接方程。对于多代数曲面,拼接过程可以考虑为不同种的连续映射。由此得到三种不同的曲面拼接方法,即串接法、过渡法和提升法。串接法可得到较低次的拼接曲面,但适用于代数曲面两两拼接且过渡曲面不相交的情况;过渡法适用于所有情况,但得到拼接曲面比较复杂;提升法是一种较好的算法,拼接时逐个将代数曲面并入拼接曲面中。该算法既可得到最低次拼接方程又适用于一般情况。上述方法的优点是无需考虑代数曲面方程中的变元,仅考虑对新增单位区间变元的处理。因此,算法的计算量小,并且能够预先得到拼接曲面时的计算量。

Three blending methods which are based on resultant theory and homotopy method are presented.Blending equations on algebraic surfaces and their auxiliary surface are constructed at first.In fact,computing blending surfaces is the procedure of elimination.Therefore,resultant method is used to blend different algebraic surfaces because of its high effieieney.These three dif- ferent methods are serial blending method,direct blending method and algebraic surface lifting blending method,for blending three or more surfaces.In these three methods,the first is only suitable for blending two surfaces in turn and there is no intersection between any two different blending surfaces;as to the second method,the blending result is complicated though it is a general al- gorithm;the last one is a good method,because it is general and the degree of blending surfaee is much lower than the seeond method.All of these methods are based on resultant theory.Therefore,the blending surface are obtained after eliminating the new added variables in unit interval and don＇t care about the original variables in algebraic equations.Beeause of these,the complexity of algorithm is low＇.Moreover,the times of computation can be defined in advance when blending algebraic surfaces.