2002 Fiscal Year Final Research Report Summary
A Comparative Study between the Conventional Euclidean Approach and the Totally 4-Dimensional Approach
| Project/Area Number |
12680400
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Intelligent informatics
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| Research Institution | Waseda University |
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Principal Investigator |
YAMAGUCHI Fujio School of Science and Engineering, Professor, 理工学部, 教授 (50117298)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHIDA Norimasa Dept. of Engineering, Tokyo University of Agriculture and Technology, Research Associate, 工学部, 助手 (70277846)
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| Project Period (FY) |
2000 – 2002
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| Keywords | CAD / CAGD / geometric modeling / computational geometry / 4-D processing |
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| Research Abstract |
A present CAD system is considered to have some deficiencies. Namely, it has many elements that are inaccurate or non-robust, and that the system itself has become too complex. These problems have a direct bearing on a system's reliability. Over the author's course of study, he has come to realize that the major cause of such inherent problems of Euclidean Geometric Processing (EGP) lies in performing division operations, and thus he proposes "Totally Four-dimensional Geometric Processing (TFGP)," which enables us to dispense with the detrimental operations.
The present research is a theoretical and experimental comparison between EGP and TFGP.
(1') Exactness: It can be said that TFGP can perform exact computations as long as it deals with rational numbers. EGP, on the other hand, is obliged to deal with approximated data which are the results of division operations inherent in EGP. This was confirmed through various experimental results.
(2') Robustness: In TFGP, there is no such instabi
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lity as division by zero, because a division operation is not ordinarily performed except at the very end of whole process. While the geometric Newton-Raphson method in EGP occasionally fails when it is applied to rational polynomial curves, i.e., the parameter value diverges and finally halts the algorithm, the TFGP method does not show such non-robustness because it treats a homogeneous curve which is expressed as an ordinary curve of dimension higher by one. This superiority is borne out by many experimental data.
(3') Compactness: A Euclidean geometry tends to be more complicated because it is considered to be a cut of homogeneous one (i.e., linear subspace) of which the dimension is higher by one than that of its counterpart. The increase of geometric types in EGP makes it much more complex. Each type must be mathematically represented individually, and the increased numbers of the combinations must be processed.
As seen above, TFGP is superior to EGP in the above three items and also in terms of generality, unifiability and duality. Less
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Research Products
(14 results)
