Minkowski sums of point sets defined by inequalities
The existing approaches support Minkowski sums for the boundary, set-theoretic, and ray representations of solids. In this paper, we consider the Minkowski sum operation in the context of geometric modeling using real functions. The problem is to find a real function f(3)(X) for the Minkowski sum of...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2003 |
| País: | México |
| Institución: | Universidad Nacional Autónoma de México |
| Repositorio: | Sistema de Información de la Facultad de Ciencias, UNAM |
| OAI Identifier: | oai:repositorio.fciencias.unam.mx:11154/1720 |
| Acceso en línea: | http://hdl.handle.net/11154/1720 |
| Access Level: | acceso abierto |
| Palabra clave: | Computer Science, Interdisciplinary Applications Mathematics, Applied shape modeling function representation Minkowski sum projection R-function |
| Sumario: | The existing approaches support Minkowski sums for the boundary, set-theoretic, and ray representations of solids. In this paper, we consider the Minkowski sum operation in the context of geometric modeling using real functions. The problem is to find a real function f(3)(X) for the Minkowski sum of two objects defined by the inequalities f(1)(X) greater than or equal to 0 and f(2) (X) greater than or equal to 0. We represent the Minkowski sum as a composition of other operations: the Cartesian product, resulting in a higher-dimensional object, and a mapping to the original space. The Cartesian product is realized as an intersection in the higher-dimensional space, using an R-function. The mapping projects the resulting object along n coordinate axes, where n is the dimension of the original space. We discuss the properties of the resulting function and the problems of analytic and numeric implementation, especially for the projection operation. Finally, we apply Minkowski sums to implement offsetting and metamorphosis between set-theoretic solids with curvilinear boundaries. (C) 2003 Elsevier Science Ltd. All rights reserved. |
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