| Surface analysis tool and its analysis capability are getting more and more respected as surface analysis is getting more and more important in the areas of computer aided geometry design and computer aided manufacturing, and its applications more and more spread in the areas such as cutter optimal positioning, facial feature matching, cerebrum analysis, animation design, and robot programming.This paper briefly summarized the theories, techniques and applications of free-form parameterized surface shape analysis. It introduced differential surface geometry as the theoretical base of free-form parameterized surface shape analysis, summed up the analysis techniques, analyzed the characteristics of the surface analysis tools in some well-known CAD software packages, and showed their lack of capabilities in globally analyzing surface shapes. Further more, the paper studied the definition of convex surface, the classification of convexity conditions, and the topological decomposition algorithm of free-form parameterized surface shape analysis. Details follow:First, we studied many convex surface definitions in history, analyzed the differences of the definitions because of the ways they are defined.Based on the previous work of other researchers, we presented the definition of global convex: a surface is globally convex if it happens to be the surface of a convex body or part of the surface of a convex body.Based on this definition, we analyzed the connotation and objection of each definition currently still in use, clarified the relationships of the definitions, and also clarified the equivalence and unequivalence to the definitions of global convex and local convex. Each definition of convex surface is restored to its original.Second, we evaluated the convexity conditions of special parameterized surface, classified general convexity conditions into geometric class and algebraic class, pointed out what we need to pay attention to when using convexity conditions algebraically. We questioned the sufficient conditions of convexity of regular parameterized surface previously put forward by other researchers and presented a counterexample. A necessary foundation is laid out for analyzing the global shape of a surface and designing new algorithms.Third, we studied the tools used to present topological information of global shape of free-form surface, analyzed the algorithm for building contour lines and its shortcoming, for the first time, we presented the GGM(Global Gauss Map) algorithm to approach the boundary of Gaussian map with arbitrary precision using quad-tree mesh decomposition method(piecewise linear polygon approximation). In this algorithm, Gaussian map is used to globally analyze the shape of a surface and complex surface is divided into single-type areas. By using this algorithm to solve the linear polynomials which represent Gaussian parabolas, calculation formulae are simplified, computation complexity is reduced, and the difficulty of the IPP algorithm by Takashi Maekawa and N.M.Patrikalakis is overcome.GGM algorithm is consisted of the computation of the boundary outside Gaussian map and the test of inclusiveness of a closed domain.We improved the original algorithm to compute the boundary outside of Gaussian map, suggested to tell the moving direction of the boundary outside of Gaussian map with the combination of both first derivatives and second derivatives, and solved the problem that tangency is out of the consideration by the original algorithm. We designed the algorithm which can perform the closed domain inclusion test in real time, realized quick search in the algebraic inclusion test algorithm by transforming polynomials with dichotomy. A real computation on a cubic Bezier surface is performed and the result of the simulation showed the new algorithm is faster than the old one by about ten fold.Finally, the paper summarized our research work, pointed out the key areas for future research, and discussed the application prospect of Gauss map computation in many fields.
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