Geometric Information and Rational Parametrization of Nonsingular Cubic Blending Surfaces

Journal of Applied Mathematics ◽

10.1155/2011/349315 ◽

2011 ◽

Vol 2011 ◽

pp. 1-16

Author(s):

Minghao Guo ◽

Tieru Wu ◽

Shugong Zhang

Keyword(s):

Symbolic Computation ◽

Parametric Representation ◽

Geometric Information ◽

Geometric Properties ◽

Rational Parametrization ◽

Cubic Surfaces ◽

Projective Transform

The techniques for parametrizing nonsingular cubic surfaces have shown to be of great interest in recent years. This paper is devoted to the rational parametrization of nonsingular cubic blending surfaces. We claim that these nonsingular cubic blending surfaces can be parametrized using the symbolic computation due to their excellent geometric properties. Especially for the specific forms of these surfaces, we conclude that they must be , , or surfaces, and a criterion is given for deciding their surface types. Besides, using the algorithm proposed by Berry and Patterson in 2001, we obtain the uniform rational parametric representation of these specific forms. It should be emphasized that our results in this paper are invariant under any nonsingular real projective transform. Two explicit examples are presented at the end of this paper.

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Look at it This Way

Proceedings of the Institution of Mechanical Engineers Part B Management and engineering manufacture ◽

10.1243/pime_proc_1986_200_048_02 ◽

1986 ◽

Vol 200 (1) ◽

pp. 55-64

Author(s):

P Cooley

Keyword(s):

Finite Element ◽

Mesh Generation ◽

Geometric Model ◽

Finite Element Mesh ◽

Solid Object ◽

Geometric Information ◽

Element Mesh ◽

Geometric Properties ◽

Complete Representation ◽

Solid Objects

A geometric model is, by definition, an informationally complete representation of a solid object. Such models are processed for various purposes: pictorial display, finite element mesh generation, geometric properties, etc. The ability to define and process representations of solid objects has enormous potential for engineers, provided that software is available which is capable of making routine decisions on such matters as how best to display the model. This paper concentrates on one set of decisions concerning the display and addresses the question of which views of the object are appropriate for a user seeking precise geometric information.

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A Boundary-Blending Method for the Parametrization of 2D Surfaces With Highly Irregular Boundaries

Journal of Mechanical Design ◽

10.1115/1.1667912 ◽

2004 ◽

Vol 126 (2) ◽

pp. 327-335 ◽

Cited By ~ 6

Author(s):

Jui-Jen Chuang ◽

Daniel C. H. Yang

Keyword(s):

Dual Process ◽

Fine Tuning ◽

New Method ◽

Laplace Method ◽

Distribution Problem ◽

Geometric Information ◽

Geometric Properties ◽

Blending Method ◽

The Given ◽

Boundary Curves

In this paper three methods are used to generate boundary-conformed parametrization of 2D surfaces. They include two conventional approaches, the Coons method and the Laplace method, and a new method, called “boundary-blending method.” In this new method, unidirectional 2D parametrization is achieved based on the geometric information of the given boundary curves. A dual offsetting procedure is adopted. The geometric properties considered for offsettings include position, curvature, and normal of the two facing parent curves. The algorithm contains two adjustable parameters that enable fine-tuning of this parametrization. This unidirectional process can be easily extended to bi-directional parametrization via superposition to include both boundary pairs. Examples show that this algorithm leads to reasonable smooth blending of the boundaries, and the dual process achieves seamless converging at the middle. It is more robust than the Coons method with regard to parametrization anomalies and relieves the relatively large uneven grid distribution problem experienced in the Laplace method. We believe that this method provides a useful alternative for the 2D boundary-conformed parametrization problems.

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A stratification of the space of cubic surfaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100056863 ◽

1980 ◽

Vol 87 (3) ◽

pp. 427-441 ◽

Cited By ~ 7

Author(s):

J. W. Bruce

Keyword(s):

Vector Space ◽

Geometric Properties ◽

Cubic Surfaces ◽

Complex Projective

In (4) the classification of (complex, projective) cubic surfaces by the number and nature of their singularities is carried out. This gives a natural partition of the vector space of cubic surfaces (which we denote by H3(4, 1)). In this paper we investigate the differential geometric properties of this partition; we show that it provides a finite constructible stratification of H3(4,1) which, in the notation of (10), is Whitney (A) regular. In fact Whitney (B) regularity holds over each stratum other than E6, but this stratum of cubic cones has an exceptional (equianharmonic) orbit at which (B) regularity fails. It remains to be seen whether or not this is the only exceptional orbit.

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On the Shape of Field-Ion Emitters

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100092566 ◽

1976 ◽

Vol 34 ◽

pp. 520-521

Author(s):

H.C. Eaton ◽

B.N. Ranganathan ◽

T.W. Burwinkle ◽

R. J. Bayuzick ◽

J.J. Hren

Keyword(s):

Grain Boundary ◽

Spherical Shape ◽

Theoretical Strength ◽

Evaporation Process ◽

Field Evaporation ◽

Geometric Information ◽

Field Ion Microscopy ◽

Considerable Error ◽

True Shape ◽

Ion Microscopy

The shape of the emitter is of cardinal importance to field-ion microscopy. First, the field evaporation process itself is closely related to the initial tip shape. Secondly, the imaging stress, which is near the theoretical strength of the material and intrinsic to the imaging process, cannot be characterized without knowledge of the emitter shape. Finally, the problem of obtaining quantitative geometric information from the micrograph cannot be solved without knowing the shape. Previously published grain-boundary topographies were obtained employing an assumption of a spherical shape (1). The present investigation shows that the true shape deviates as much as 100 Å from sphericity and boundary reconstructions contain considerable error as a result.Our present procedures for obtaining tip shape may be summarized as follows. An empirical projection, D=f(θ), is obtained by digitizing the positions of poles on a field-ion micrograph.

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