二次代数曲面拼接中的光顺处理

针对直接采用理想交理论得到的拼接曲面在实际中不一定连续的问题,研究如何通过改变拼接曲面的构造方程以得到连续的拼接曲面及其光顺处理。首先,分析了拼接曲面在实际应用不连续的原因,若过渡曲面中含某个变元的项在其他变元满足某个值时变为0,则其与该变元不再相关,在几何图形上会表现为断开;然后,给出了保证拼接曲面在实际应用中连续的方法;之后,讨论了0阶和任意阶拼接曲面的光顺处理方法。对于0阶光滑连续曲面,将辅助曲面作为因子乘以一次函数后补偿到构造方程中的主曲面部分,调节参数使得拼接曲面光顺;对于任意阶连续曲面,在主曲面过渡方程中直接增加补偿函数。该方法可使0阶光滑连续曲面在不提高次数的情况下做到光顺。

Smoothening in surface blending of quadric algebraic surfaces

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.