| The implicitization of rational surface is a classic algebraic geometry problem,essentially it's an elimination issue,which has important applications in Geometric Modeling and Computer Aided Geometric Design.Though we have proved theoretically that every rational parametric equation can be implicitized successfully,currently every implicitization method has its limitations in the industry.The algorithmic complexity of Gr(?)bner basis method is too complicated;The algorithmic complexity of Wu-method usually has been reduced,but it's still difficult to meet the high performance requirements;The Resultant-based method can't work well when the rational surface with base points;The moving surfaces method created by Sederberg and Chen in 1995 is the most efficient way for the cases without base points,however it need to be further researched when the base points exist.So the implicitization problem has not been completely solved and it's still a hot topic for researchers.In reality,many rational surfaces are sparse,so research on implicitization of them has practical significance.This paper try to build a new implicitization way for sparse rational surfaces.The main contents include:Based on the rational surface,use the integer programming to solve the best weight set and build the Gale duality equations,then use the Resultant to elimination,and get the implicit equation by polynomial factorization;Prove the algorithm by Polynomial,Varieties,Ideals,Resultant,Gale Duality,etc theories;Do complexity analysis with the other implicitization methods.From the research results:The performance of the method build in this paper has been increased significantly in some cases compared with the Gr(?)bner basis and Wu-method,so it can solve many implicitization problems which can't be solved well by Gr(?)bner basis and Wumethod.The method in this paper works even the rational surface with base points,also it works well whether the rational surface is proper or not,so it has widely applications. |
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