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• W2413703831 abstract "To aid computer programming for geometric applications such as CAD/CAM, computer graphics and robot control, it is necessary to equip geometric programming languages with special data types that can represent the Euclidean transformations such as rotations and translations and the geometric entities such as points, vectors and coordinate frames existing in the three-dimensional Cartesian and Euclidean spaces. This thesis was written to address two fundamental problems pertaining to the specification and the implementation of these geometric data types.The first problem is concerned with the representation of three-dimensional rigid-body rotations using finite number of digits. The finite-precision representation of rotations currently used all possess certain mathematical defects: they may lose their length-preserving property due to rounding of their parameters, or have singularities at which the representations degenerate, or lack the formulae for performing multiplication of rotations in terms of the representations. Aiming at supporting robust geometric computation, we developed a family of isometric finite-precision rotation representations by covering the rotation groups with analytic coordinates charts that are left-translated by the elements of finite rotation subgroups. Among them, the representation based on four Cayley's coordinate charts offers both efficient algorithms for rotation operations and small errors due to parameter rounding. We recommend to use this representation to replace the conventional representations used in geometric computation.The second problem is concerned with data encapsulation in the geometric data types. Geometric entities are specified in a relative fashion: their numerical values are meaningful only if their reference coordinate frames are specified and a geometric computation can be performed only when all the operands are specified with respect to the same reference frame. Aiming at supporting object-oriented geometric programming, we define two families of abstract data types. The first family, referred to as value types, encapsulates the finite-precision representation of Euclidean transformations. The second family, referred to as entity types, encapsulates a value-type object and a reference frame, provide the abstraction of the geometric entities such as reference frames, points, and vectors existing in the Cartesian and Euclidean spaces. Geometric entities specified by entity types can be referred and manipulated without concern for their reference frames on the part of the programmers.Aside from theoretical investigation, we implemented a twenty-four chart Cayley's representation of rotations and selected value and entity types as user-defined classes in C++. We also demonstrated the use of these data types in a simple graphic animation program." @default.
• W2413703831 created "2016-06-24" @default.
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• W2413703831 date "1996-01-01" @default.
• W2413703831 modified "2023-10-16" @default.
• W2413703831 title "Finite-precision representation and data abstraction for three-dimensional euclidean transformations" @default.
• W2413703831 hasPublicationYear "1996" @default.
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