J. ROSSIGNAC AND A. REQUICHA, Offsetting operations in solid modeling, Computer-Aided Geometric Design 3 (1986) 129-148.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....data structures and the elegant recursive algorithms further contributed to the popularity of CSG in academia and early commercial systems. 3.1.3. Other constructive representations In principle, many other constructions may be added to the lexicon of implicit representations, notably offsetting [92], blending [133] convolutions [11] and other skeletal based representations, Minkowski operations [26, 62] and sweeping [37, 114] Such constructions have numerous applications in mechanical design, analysis, and planning tasks; they also flourished in computer graphics [112] where ....

J. R. Rossignac and A. A. G. Requicha. Offsetting operations in solid modeling. Computer Aided Geometric Design, 3(2):129--148, 1986.

....principal directions are the corresponding eigenvectors p # and p # , which are the columns of the matrix P . Now consider that a spherical robot finger is rolling and sliding over the surface of an object with point contact at all times. The center point of the finger generates an offset surface [42]. The concept of a parallel surface can be used to develop a description of the offset surface traced by the fingertip s center of curvature. A parallel surface is constructed by drawing a normal line (determined by the z axis of the Gauss frame, z#u#) with length r # at each point on the surface. ....

J. R. Rossignac and A. A. G. Requicha. Offsetting operations in solid modeling. Computer Aided Geometric Design, 3(2):129--148, 1986.

....funding agencies. 4 The research of the fourth author was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. Andersson, Dorney, Peters, Stewart: Polyhedral Perturbations 2 1 Introduction In many applied fields, including tolerancing and metrology [1, 2, 3, 4, 5, 6, 7, 8, 9], solid modeling [10, p.110] engineering design [11] finite element analysis [12] surface reconstruction [13] computer graphics [14] path planning in robotics [15, p. 91] fairing procedures [16] image analysis [17, p. 69] and medical imaging [18, 19] it is natural to require that a ....

Rossignac, J. R. and Requicha, A. A. G. (1986) Offsetting operations in solid modeling, Computer Aided Geometric Design 3, pp. 129-148.

....operation which expands a given object into a similar object by a certain extent. It has various important applications in tolerance analysis, cutter path generation for NC machine tools, collision free path planning for robot motions, and for constant radius rounding and filleting of solids [20]. Constant radius offsetting for plane curves has attracted extensive research interests in computer aided design [7, 8, 9, 13, 14, 15, 18, 22] Since exact offset curves have high degree curve equations which are computationally quite expensive [8, 9, 13] many authors have considered ....

Rossignac, J. and Requicha, A., "Offsetting Operations in Solid Modeling," Computer Aided Geometric Design, Vol. 3, pp. 129--148, 1986.

.... Offset and (b) convolution of planar objects Offset and convolution computations are classic operations in CAD CAM, which can be used in various interesting geometric applications such as NC machining [8, 18] motion planning [3, 17, 28] character font and brush stroke design [14, 15] blending [31], and shape transformation [22] The exact offset and convolution curves of planar algebraic curves are algebraic, but not rational. Moreover, they have very high algebraic degree [12, 19, 23] For example, the offset of a cubic B ezier curve has an algebraic degree of 10 [12] Consequently, ....

....Methods 3. 1 Planar Convolution Curve Convolution is a classic operation which has been used as a tool for computing collision free paths in robot motion planning [3, 17, 28] Moreover, the convolution operation has applications in character font design [14, 15] offset and rounding [31], and shape transformation [22] a) b) c) d) C1 C2 C 2 C 1 x y GammaC 2 C 1 C 1 ( GammaC 2 ) Fig. 3.1. Convolution and C space obstacle Given two regular parametric curves: C 1 (t) x 1 (t) y 1 (t) t 0 t t 1 , and C 2 (s) x 2 (s) y 2 (s) s 0 s s 1 , we define ....

Rossignac, J.R., and Requicha, A.A.G. (1986): Offsetting operations in solid modeling. Computer Aided Geometric Design, 3:129--148.

....S d be the offsets of T and S, respectively, with respect to the offset distance d. When the offset surfaces T d and S d intersect in a degenerate circle of radius R, the torus T and the sphere S can be blended using a torus with a major radius R and a minor radius d (see Rossignac and Requicha [25, 26]) The rest of this paper is organized as follows. In Section 2, we define some basic notations and review mathematical preliminaries. Section 3 presents geometric algorithms to compute the TSI curve. Finally, we conclude this paper in Section 4. 5 (a) b) c) d) S T S T S T S T ....

Rossignac, J., and Requicha, A., "Offsetting Operations in Solid Modeling," Computer Aided Geometric Design, Vol. 3, No. 2, pp. 129--148, 1986.

....we consider the difficulties in computing constant radius offsetting, which is a special case of convolution for which one operand is simply a circle of fixed radius r. With its important application to NC tool path generation, the offsetting problem has attracted much attentions from many authors [EC91, FN90, Hos85, RR86]. The approach to compute the exact offset curve equations, however, has limitations in practice. It is because the exact offset curves have very high degrees. For example, the offset curve for a B ezier cubic curve may have degree 10 in general [FN90] Hoffmann [Hof89a, Hof89b] suggests an offset ....

....only one intersection. Thus, the total computation time is bounded by O(h) and the time for local refinments. 3. 3 Point Classification The problem to determine whether a point is in the interior, in the exterior, or on the boundary of an object is very important in various geometric decisions [Hof89a, RR86, Til80]. This problem can be reduced to that of testing a point with respect to the boundary curves of the object. For a given point p 0 = x 0 ; y 0 ) consider the boundary curves C which intersect at p C = x C ; y 0 ) with the horizontal line y = y 0 . For a curve C with its y coordinates increasing ....

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Rossignac, J.R., Requicha, A.A.G., (1986), "Offsetting Operations in Solid Modeling," Computer Aided Geometric Design, Vol. 3, pp. 129--148.

.... Offset and (b) convolution of planar objects Offset and convolution computations are classic operations in CAD CAM, which can be used in various interesting geometric applications such as NC machining [8, 17] motion planning [3, 16, 27] character font and brush stroke design [13, 14] blending [30], and shape transformation [21] The exact offset and convolution curves of planar algebraic curves are algebraic, but not rational. Moreover, they have very high algebraic degree [12, 18, 22] For example, the offset of a cubic B ezier curve has an algebraic degree of 10 [12] Consequently, these ....

....Methods 3. 1 Planar Convolution Curve Convolution is a classic operation which has been used as a tool for computing collision free paths in robot motion planning [3, 16, 27] Moreover, the convolution operation has applications in character font design [13, 14] offset and rounding [30], and shape transformation [21] a) b) c) d) C 1 C 2 C 2 C 1 x y GammaC 2 C 1 C 1 ( GammaC 2 ) Fig. 8. Convolution and C space obstacle Given two regular parametric curves: C 1 (t) x 1 (t) y 1 (t) t 0 t t 1 , and C 2 (s) x 2 (s) y 2 (s) s 0 s s 1 , we define their ....

Rossignac, J.R., and Requicha, A.A.G.: Offsetting operations in solid modeling. Computer Aided Geometric Design, 3 (1986) 129--148

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J. ROSSIGNAC AND A. REQUICHA, Offsetting operations in solid modeling, Computer-Aided Geometric Design 3 (1986) 129-148.

....not be computed by simply considering distances between vertices and edges of one set and the other set. To illustrate this point, consider the equivalent definition of H(O,A(V) as the minimum r for which O # A(V ) # r and A(V ) #O # r,whereX # r is the offset of the set X by a distance r [25] or equivalently is the Minkowski sum of X with a ball of radius r centered at the origin [28] The maximum deviation may happen at a point c in the interior of a face of O , such that the open ball of center c and radius less than r does not intersect A. Because the exact Hausdorff distance is ....

J. Rossignac, A. Requicha, Offsetting operations in solid modeling, Comput. Aided Geom. Design 3 (2) (1986) 129--148.

....x2 B x2 B cells and snapping each vertex to the center of the nearest corner of its cell. The simplest way of constructing a simplified version of a triangle mesh O is to perform the quantization described above and to remove degenerate triangles, which have at least two coincident vertices [20]. Variations on this approach and more complex, yet more effective, simplification techniques are reviewed in [23] Many of these techniques [12,18,5,6] simplify the model incrementally by collapsing one edge at a time and by discarding the triangles that become degenerate. 1.2 Problem Let O be ....

....the analysis of the sphere, and Section 6 gives the actual formulas for computing B, V, E, and F. Section 7 outlines the algorithm used to estimate K, and Section 8 presents our empirical results. 2. Prior Art 2.1. Optimal bit allocation Bit quantization was used for mesh simplification [20] and for compression [3,12,30,8,34] but the number of bits used for compression was selected by the operator, through visual criteria and trial anderror. Chow [2] provides an algorithm for selecting the quantization level, based on testing each triangle in a model against the size of the ....

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J. Rossignac & A. Requicha, "Offsetting Operations in Solid Modeling," Comput. Aided Geom. Design 3, No.2, 129-148, August 1986.

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J. Rossignac and A. Requicha. Offsetting operations in solid modeling. Comput. Aided Geom. Design, 3:129--148, 1986.

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Rossignac, J. R. and Requicha, A. A. G., "Offsetting operations in solid modeling," Computer Aided Geometric Design, 3, pp. 129--148, 1986.

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J. Rossignac, A. Requicha (1984), "Offsetting operations in solid modeling," CAGD 3, 129--148.

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