Abstract: As many geometric modelings essentially are the problems of the surface blending with constrained conditions, a nonlinear homotopy mapping method was presented to compute the surface equation of three-dimensional modeling on the base of linear homotopy method. In the method, the interpolation polynomial was computed firstly by using the position of cross-over section or biological slices as the interpolation points. Then, this interpolation polynomial was regarded as the nonlinear continuous homotopy mapping function and substituted into the polynomials of primary surfaces and auxiliary surfaces respectively to get blending surface equation. Thus, two univariable equations were obtained when the interpolated variable in interpolation polynomial was used as the variable but the others in the equations of primary surfaces and auxiliary surfaces were used as parameters. Furtherly, Sylvester resultant was used to eliminate the interpolated variable in these two equations to achieve the modeling surface which satisfied the constraints. The proposed method can realize surface modeling with control points and geometric modeling with constraints, and it is more practical because it can redefine and change the the intermediate position and shape.

Key words: nonlinear continuous homotopy mapping, algebraic surface, resultant, surface blending, geometric modeling

摘要： 针对很多几何造型是带有约束条件的曲面拼接问题,在线性连续同伦的基础上提出了利用非线性同伦连续计算拼接曲面以进行三维造型的方法。首先,根据得到的截面(切片)的位置及其曲线方程确定插值点并得到插值多项式;其次,将此插值多项式作为非线性连续同伦映射函数并分别代入主曲面和辅助曲面的多项式方程得到过渡曲面的方程;然后,仅将插值变元作为变元而主、辅助曲面方程的变元作为参数,利用Sylvester结式消去过渡方程中的变元得到关于主曲面的拼接方程即造型曲面。利用该方法能实现带有控制点的曲面造型以及多曲面约束的几何造型,而且它可以确定造型过程中的中间形状及中间形状的位置,从而更加具有实用性。

关键词: 非线性同伦映射, 代数曲面, 结式, 曲面拼接, 几何造型