Research On Geometric Modeling Simplification Technology In CAD

It is better to represent geometric model with less amount of data by modeling tools in CAD modeling systems. This is good for reducing computing time, improving the efficiency of systems, speeding up the network transmission speed. Bézier curves, surfaces and B-spline are widely used as modeling tools in CAD/CAM. Their models' simplification technology has practical significance on the CAD systems. In this paper, we have made in-depth studies on simplification technology, mainly two aspects: approximate merging of Bézier curves, surfaces and multi-degree B-spline. The main creative results are as follows.Firstly, in constrast to traditional methods, which only considered the components of the curves separately, we used geometric information about the curves to generate the merged curve and proposed optimal approximate merging of a pair of Bézier curves with G²-continuity in L₂ norm, where positions, tangents and curvatures were preserved at the two endpoints. For avoiding singular at the two endpoints, we amended the error definition and added one regularization term. Compared to traditional methods, our method could directly obtain control points of the merged curve, regardless of the degrees of the original curves and the approximation error was better. Furthermore, we obtained a higher degree merged curve through raising the merged Bézier curve's degree instead of degree elevation of the original Bézier curves.Secondly, approximate merging of two adjacent tensor product Bézier surfaces was investigated to guarantee the compression of geometric data in CAD systems. Sufficient and necessary condition for precise merging of adjacent tensor product surfaces was obtained by using the matrix representation of subdivided Bézier surface, then merged tensor product Bézier surface was solved by generalized inverse matrices in L₂ norm based on precise merging condition and explicit representation of the merged surface's control points was also obtained. At the same time, the result of approximate merging with corner interpolation was showed first time. Since the minimal least squares solution could be directly obtained by generalized inverse matrics, an approximate merging algorithm possessing explicit formula, less time consumption and better approximation result was found.After that, multi-degree B-spline(MD-spline) was investigated. Multi-degree B-spline is a new B-spline form to simplify geometric model and compress the amount of data. This paper made a preliminary study, and basis function formulae of MD-splines which maximal variational degree was lower than 3 was investigated first time. We gave basis function's properties and curve's construction completely, then applied them to degree elevation and mergence of spline curves for simplifing curve model. After degree elevation of algebraic hyperbolic B-spline can be interpreted as corner cutting using multi-degree B-spline successfully, we obtained geometric construction of algebraic hyperbolic B-spline based on above conclusion. Since algebraic hyperbolic B-spline has important meaning for modeling system, for example, it can represent hyperbola, catenary explicitly etc common engineering curve, this algorithm processes practical application value. Similarly, we can make use of multi-degree spline to obtain geometric construction of other splines, such as NUAT B-spline.Finally, in view of the previous triangle or hyperbolic polynomial spline model at uniform knots on the definition of the defect, taking a algebraic hyperbolic blending B-spline as an example, this paper extended to non-uniform knots, and the new spline holds a lot of good geometric properties such as B-basis.