W Boehm, G Farin, and J Kahmann. A Survey of Curve and Surface Methods in CaGD. CAGD, 1:1--60, 1984. 284  Home/Search   Document Not in Database   Summary   Related Articles   Check

....schemes, their properties and recent research on high dimensional subdivision algorithms. 2.1 Parametric Representation In the beginning of the computer graphics, parametric representation has been one of the most important technique to represent geometric objects. In 1984, Boehm et al.[17] surveyed parametric curves and surfaces which had been widely used especially in computer aided design and manufacturing. Basically, we express an object as a mapping from a domain to three dimensional Euclidean space. y = f(x) y (13) where x , for some domain D. Obviously, k depends ....

W. Boehm, G. Farin, and J. Kahmann. A survey of curve and surface methods in CAGD. Computer Aided Geometric Design, 1:1--60, 1984.

....surface from a net of control points [73, p. 18] and Ferguson gave a method for interpolating vector valued surface data by composite bicubic surfaces [93] B ezier surfaces were introduced in the late sixties and they still remain, together with B splines surfaces, one of the most popular schemes [26]. The triangular polynomial patches of de Casteljau [72, p.10] are widely used in the nite elements method. Radial basis functions, which are now very much in avor, can be considered as a generalization of splines to several dimensions [204] Many historical notes on splines can be found in ....

W. Bohm, G. Farin, J. Kahmann, A survey of curve and surface methods in CAGD, Comput. Aided Geom. Des., 1 (1984) 1-60.

....entire patch. In the former case triangular or rectangular elements are put together [2,6,12,20,23] or recursive subdivision methods are applied [5,8,24] In the latter case either the known control point based methods are generalized or a weighted sum of 3D interpolants gives the surface equation [1,3,4,22,26]. The method presented in this paper is a recursive subdivision scheme specially designed to consider arbitrary boundary conditions. Subdivision schemes provide efficient algorithms for the design, representation and processing of smooth surfaces of arbitrary topological type. Their simplicity ....

W. Boehm, G. Farin, and J. Kahmann. A survey of curves and surface methods in cagd. Computer Aided Geometric Design, 1(1):1--60, 1984.

.... In the former case triangular or rectangular elements are put together [2, 6, 12, 20, 23] or recursive subdivision methods are applied [5, 8, 24] In the latter case either the known control point based methods are generalized or a weighted sum of 3D interpolants gives the surface equation [1, 3, 4, 22, 26]. This paper presents a subdivision schemes specially designed for the task of filling n sided holes, which belongs to the former case. Subdivision schemes provide efficient algorithms for the design, representation and processing of smooth surfaces of arbitrary topological type. Their simplicity ....

W. Boehm, G. Farin, and J. Kahmann. A survey of curves and surface methods in cagd. Computer

....2.6. Let s be a piecewise polynomial function de ned on T 1 [ T 2 . Then s 2 7 S r q (fT 1 ; T 2 g) i for all = 0; r: a [T2 ] i;j; X i 1 j1 k1= a [T1 ] i i1 ;j j1 ;k 1 q i 1 j 1 k 1 i 1 1 (v 4 ) j 1 2 (v 4 ) k1 3 (v 4 ) i j = q : It is well known (cf. [4]) that in the case r = 1 the smoothness conditions of Theorem 2.6. have the following geometric interpretation: The points ( i q v 1 j q v 2 v 4 ; a [T2 ] i;j;1 ) i 1 q v 1 j q v 2 ; a [T1 ] i 1;j;0 ) i q v 1 j 1 q v 2 ; a [T1 ] i;j 1;0 ) i q v 1 j q v 2 ....

W. Boehm, G. Farin and J. Kahmann, A survey of curve and surface methods in CAGD, Comp. Aided Geom. Design 1 (1984), 1-60.

....relations show that B ezier Bernstein coe cients can be considered as certain linear functionals (see Section 3) The B ezier Bernstein representation (1) of a polynomial p T 2 q has important applications in CAGD. We refer the reader to the surveys of Farin [71] Boehm, Farin, and Kahmann [21], the book by Chui [31] and the papers of de Boor [24] and Dahmen [49] Moreover, concerning the so called blossoming approach, we refer to the tutorial of de Rose, Goldman, and Lounsbery [139] and the survey of Seidel [154] Triangulation methods were described in the survey of Schumaker [152] ....

W. Boehm, G. Farin, and J. Kahmann, A survey of curve and surface methods in CAGD, Comp. Aided Geom. Design 1 (1984) 1-60.

....convex quadrangulation with diagonals in [46] where di erent methods are used. 3 Construction of Admissible Sets In this section, we construct admissible sets for spline spaces S r q ( where q 3 if r = 1, and q 5 if r = 2. In order to describe admissible sets we need some notations (cf. [5, 6, 9, 23, 24]) Let T [l] v [l] 1 ; v [l] 2 ; v [l] 3 ) l = 1; N , be the triangles of . For s 2 S r q ( the polynomials p [l] sj T [l] 2 q ; l = 1; N , can be written as p [l] x; y) X i j k=q a [l] i;j;k q i j k i 1 (x; y) j 2 (x; y) k 3 (x; y) ....

W. Boehm, G. Farin and J. Kahmann, A survey of curve and surface methods in CAGD, Comp. Aided Geom. Design 1 (1984) 1-60.

.... which are used for many applications in Computer Aided Design and Manufacture, such as reliable surface plotting and display, motion display (computing transformations) computing cutter o set surfaces, computing curvatures for shading and colouring, and many others (see also [BFK84, Bar85, QD87, Far88, Hof89] NURBSs are already a de facto standard in CAD CAM environments (see [Nie93, BCX94] and their popularity is still growing. 8 3.2 Solving Geometric Problems on Irrational Surfaces The scope of the method of adjoints reaches beyond parameterization. Like in the ....

W. Bohm, G. Farin, and J. Kahmann. A survey of curve and surface methods in cagd. Comp. Aided Geom. Design, 1:1-60, 1984.

....the surface case (bivariate) Such a scheme is called uniform since the coefficients of these masks are fixed. In each iteration the number of points is roughly doubled in case of curves, and is quadrupled in case of surfaces. Some BSS serve as tools for computing spline curves and surfaces (see [3], 7,8] while other useful schemes introduced in CAGD converge to non standard limits. For example see de Rahm [21] Catmull and Clark [4] Doo and Sabin [12] Dubuc [13] Deslauriers and Dubuc [11] Dyn, Gregory and Levin [15,16,17] The convergence analysis of uniform BSS has been developed ....

Boehm, W., G. Farin and J. Kahmann, A survey of curve and surface methods in CAGD, Comp. Aided Geom. Design 1 (1984), 1--60.

....subdividing polynomial triangular surface patches. Subdivision methods based on a version of the de Casteljau algorithm splitting a control net into control subnets (see Farin [5] were investigated by Goldman [11] Boehm and Farin [2] Bohm [4] and Seidel [16] see also Boehm, Farin, and Kahman [3], and Filip [8] However, except for Bohm [4] these papers are not particularly concerned with minimizing the number of calls to the standard de Casteljau algorithm. Furthermore, some of these papers (notably Goldman [11] use a version of the de Casteljau algorithm computing a 5 dimensional ....

W. Boehm, G. Farin, and J. Kahman. A survey of curve and surface methods in CAGD. Computer Aided Geometric Design, 1:1--60, 1984.

....to form an infinite periodic sequence. The centroid of the points is denoted by b : 1 n P n01 j=0 b j , and the Bernstein polynomials of degree r are fi (r) j (t) r j (1 0 t) r0j t j ; 0 j r; t 2 [0; 1] Then we consider the Bernstein B ezier polynomials [1] 2] [3] f r (t) r X j=0 b j fi (r) j (t) for large degrees r and investigate the behavior of f r (t) in the convex hull of the control points, when r tends to infinity. We want to characterize all limit points of the curves f r (t) as shown by figures 1 and 2 for n = 4 and n = 7 ....

Boehm, W., G. Farin, and J. Kahmann, A survey of curve and surface methods in CAGD, Comput. Aided Geom. Design 1 (1984) 1-60

....conditions as the original cubic curve, as can be seen in figure 2. 2.2. Previous work A large amount of work has been done in relation to the conversion from polyhedra to curved patches, which also alleviate the problem of straight silhouettes. A survey of much of this work can be found in ([3]) Most subdivision schemes that yield cross boundary continuity, however, rely on the availability of vertices of adjacent polygons (such as the Catmull Clark subdivision scheme, 4] rather than vertices and normal vectors of the polygon to be replaced proper. The requirements 1.1 and 1.2 render ....

W. Boehm, G. Farin, and J. Kahman. A survey of Curve and Surface methods in CAGD. Computer Aided Geometric Design, 1:1--60, 1984.

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W. Boehm, G. Farin, and J. Kahmann. A survey of curve and surface methods in CAGD. Computer Aided Geometric Design, 1(1):1--60, 1984.

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W Boehm, G Farin, and J Kahmann. A Survey of Curve and Surface Methods in CaGD. CAGD, 1:1--60, 1984. 284

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Bhm, W., Farin, G., and Kahmann, J., "A Survey of Curve and Surface Methods in CAGD", Computer Aided Geometric Design, Vol. 1 No. 1, July, 1984

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W. Bo hm, G. Farin, and J. Kahmann, "A survey of curve and surface methods in CAGD," Comput. Aided Des. 1,1--60#1984#.

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B ohm W., Farin G., Kahmann J., A survey of curve and surface methods in CAGD, Computer Aided Geometric Design 1 (1984), 1--60.

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Boehm, W., G. Farin and J. Kahmann, A survey of curve and surface methods in CAGD, Computer Aided Geometric Design 1 (1984), 1-60.

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W. Boehm, G. Farin and J. Kahmann, `A survey of curve and surface methods in CAGD'. Computer Aided Geometric Design 1, no. 1: pp. 1--60, (1984).

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Boehm, W., G. Farin, and J. Kahmann, A survey of curve and surface methods in CAGD, Comput. Aided Geom. Design 1 (1984), 1--60.

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W. Boehm, G. Farin, and J. Kahmann. A survey of curve and surface methods in CAGD. Computer-Aided Geom. Design, 1:1--60, 1984.

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Boehm, W., Farin, G.F., Kahmann, J. (1984), A survey of curve and surface methods in CAGD, Computer Aided Geometric Design 1, 1 -- 60

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W. Boehm, G. Farin, and J. Kahmann. A survey of curve and surface methods in CAGD. Computer-Aided Geom. Design, 1:1--60, 1984.

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W. Boehm, G. Farin and J. Kahmann, `A survey of curve and surface methods in CAGD'. Computer Aided Geometric Design 1, no. 1: pp. 1--60, (1984).

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Boehm (eds.), North Holland, 77--91. Boehm, W., G. Farin, J. Kahmann (1984), A survey of curve and surface methods in CAGD, CAGD 1, p. 43.

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