1988 Fiscal Year Final Research Report Summary
Study on Representations and Processing of Geometric Objects in Terms of Mutual Constraints
| Project/Area Number |
62580017
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
Informatics
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| Research Institution | University of Tokyo |
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Principal Investigator |
SUGIHARA Kokichi Faculty of Engineering, University of Tokyo, 工学部, 助教授 (40144117)
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| Project Period (FY) |
1987 – 1988
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| Keywords | Mutual Constraints / Geometric Algorithm / Shape Representation / Degeneracy Avoidance / Numerical Stabilization / Computer Vision / Voronoi Diagram Construction / 陰線消去 |
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| Research Abstract |
The possibility and the use of representing geometric objects, not in terms of coordinate systems, but in terms of mutual constraints among elements such as points, lines and faces of the objects was considered from an engineering point of view, and the following results have been obtained.
1. The mathematical structure of dependence among mutual constraints such as lenghts between vertices and angles between faces was :clarified; the dependence structure was characterized by some combinatorial properties of the geometric objects for the case where geometric elements were in general positions, whereas the dependence structure was characterized by algebraic properties for the case where geometric elements were in some special position so that degeneracy took place. 2. The above result was applied to the problem of finding inconsistency in the definition of geometric structures, to that of locating a mobile robot with a single camera, and to that of recognizing objects extracted from range pictures. 3. A new method was constructed for avoiding geometric degeneracy. Geometric degeneracy takes place when geometric elements come to some special position, and it causes to make geometric algorithms complicated. A technique called "symbolic perturbation" was applied to avoidance of such degeneracy in Voronoi diagram construction and in hidden line elimination. 4. A new method was proposed for making geometric algorithms stable against numerical errors. The method is based on the principle that the highest priority is placed on the consistency of the topological structures if numerical results contradict topological structures. This method was applied to the construction of Voronoi diagrams, and experimental results showed the validity of the method.
The new concepts and methods established in this projects will be applied to many other geometric problems in various fields of engineering in future.
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