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.

To solve the problem of discontinuity when blending two surfaces with coplanar perpendicular axis, this paper discussed how to improve the equations about the blending surface so as to obtain the smooth and continuous blending surface. At first, this paper analyzed the reason of the uncontinuousness in the blending surface and pointed out that the items in one variable were removed when other variables equaled to some specified values. In this case, the blending equation was independent to this variable in these values and this indicated that the belending surface was disconnected. Then, a method which guarantees the blending surface countinuous was presented on the basis of above discussion. Besides this, this paper discussed how to smoothen it once the continuous blending surface was computed out. As for the G0 blending surface, regarding the polynomial of auxiliary surface as a factor, this factor was mulitiplied to a function f′ with degree one and the result was added to the primary surface fi. The smoothness of blending surface can be implemented by changing the coefficients in f. For the Gn blending surface, a compensated polynomial with degree at most 2 was added to the proposed primary blending equation directly when computing blending surface. This method smoothens the blending surface but does not increase the degree of G0 blending surface.