关键词: 非线性同伦映射, 代数曲面, 结式, 曲面拼接, 几何造型

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