H. Seidel, "Polar forms for geometrically continuous spline curves of arbitrary degree", ACM Trans. Graph., 12, No.1, 1-34 (1993). 18  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that article are already known, the proofs are shorter than the previous ones and they also give further insights into the basic theory of B splines construction. Recently, blossoming has been applied to the formulation of triangular B splines surfaces [8,9] and geometric continuous spline curves [10], computation of product of splines [11] and composition of Bezier simplexes [12,13] Besides, extensions of the technique to multirational framework can be found in [14,15] In [16] Duchaineau shows that general polar values (GPV) may also be used to control a polynomial curve when a related ....

....(i.e. # spline) tensor product surfaces [21] Triangular B spline surfaces can model complex objects with non rectangular topology while # spline provides extra shape control parameters based on geometric continuity. Light has already thrown on how to construct those surfaces using blossoming [8,10] and we plan to extend our research work to those types of surfaces. Acknowledgements The author thanks reviewer s valuable comments and Prof. Stephen Mann of University of Waterloo for his help in providing the technical report [22] through the Internet. ....

Seidel HP. Polar forms for geometrically continuous spline curves of arbitrary degree. ACM Transactions on Graphics 1993;12(1):1}34.

....It is desirable to extend the basic framework of the library to allow other bases to be used. Some work has been done that relates B bases to other bases, such as convolution bases, P olya bases, and L bases [14, 13] 62 63 A datatype for geometrically continuous curves (universal splines [18]) should be supported, in the same way as the B spline datatype in Chapter 6. A datatype for tensor product surfaces need to be supported in order for the library to be generally useful. They can be implemented in a way similar to the B spline class. Finally, while a simple relationship is known ....

Hans-Peter Seidel. Polar forms for geometrically continuous spline curves of arbitrary degree. ACM Transcations on Graphics, 12(1), January 1993.

....length as the parameter. Among the various local parameterizations of the space curve, taking arc length as parameter has several advantages as indicated in section 3. In recent years, several authors have discussed the notion of geometric continuity at a common point for two incident space curves[13, 15, 24, 25]. Furthermore there are a plenty of references which use di erent continuity criteria and construct parametric B splines to approximate an ordered list of points (see for e.g. 9, 11, 25] The frame continuity used in this paper is a simpler form of geometric continuity with the connection matrix ....

.... authors have discussed the notion of geometric continuity at a common point for two incident space curves[13, 15, 24, 25] Furthermore there are a plenty of references which use di erent continuity criteria and construct parametric B splines to approximate an ordered list of points (see for e.g. [9, 11, 25]) The frame continuity used in this paper is a simpler form of geometric continuity with the connection matrix being diagonal, and di ers from the well known Frenet frame continuity that has a lower triangular connection matrix. We also exhibit how Pad e approximation can be adapted to yield very ....

Seidel, H-P., \Polar Forms for Geometrically Continuous Spline Curves of Arbitrary Degree", ACM Trans. of Graphics, 12, 1, (1993), 1 - 34. 19

....to be implemented in derived classes. At the lowest level of the hierarchy are several classes: currently UBBasis, NUBBasis, BezBasis, and CMSBasis, each individually derived from BBasis. In order they implement cardinal B splines, general B splines, Bezier splines, and connection matrix splines [4]. Each of these classes has member data related to knots: start and stride for the cardinal splines, knots themselves for the general B splines, breakpoints for the Bezier splines, and finally, breakpoints with associated connection matrices for the CMS splines. Each is responsible for ....

Seidel, H-P. Polar forms for geometrically continuous spline curves of arbitrary degree, ACM Trans. Graphics 12 (1993), 1--34.

.... and an adaptive sampling of points (x i ; y i ) on all the branches of the plane curve C: f(x; y) 0 of degree d, a possible approximation scheme is is to directly use B splines to generate piecewise rational approximations with C k Gamma1 continuity using degree k d parametric polynomials [25, 15, 22, 23, 28]. We alternatively also show here that C 1 Pad e rational approximations, with controllable approximation error, can be easily constructed since it is based on the same GCD algorithm as the one required for the curve resolution (branch separation and local parameterization) at singular points. ....

Seidel, H-P., "Polar Forms for Geometrically Continuous Spline Curves of Arbitrary Degree ", ACM Trans. of Graphics, 12, 1, (1993), 1 - 34.

....i 1 X i=n Gamma1 a (1) i (p (1) 1 )ff i 3 (4.19) Y 2 (ff 3 ) x 2 (ff 3 ) y 2 (ff 3 ) T = p (2) 2 h v (2) 1 Gamma p (2) 2 i ff 3 h p (2) 1 Gamma p (2) 2 i 1 X i=n Gamma1 a (2) i (p (2) 2 )ff i 3 (4. 20) Now we need to determine the so called fi matrix(see [29]) C (l) 2 6 6 6 6 6 4 fi 1 fi 2 fi 2 1 fi 3 3fi 1 fi 2 fi 3 1 fi 4 3fi 2 2 4fi 1 fi 3 6fi 2 1 fi 2 fi 4 1 : 3 7 7 7 7 7 5 = h fi (l) ij (p (1) 1 ) i and a (l) i (p (1) 1 ) for l = 1; 2 and i = n Gamma 1; Delta Delta Delta , so that 2 6 6 6 6 ....

Seidel, H-P. Polar Forms for Geometrically Continuous Spline Curves of Arbitrary Degree. ACM Trans. on Graphics, 12(1):1--34, 1993.

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H.-P. Seidel. Polar forms for geometrically continuous spline curves of arbitrary degree. ACM Trans. Graph., 12:1--34, 1993.

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H. Seidel, "Polar forms for geometrically continuous spline curves of arbitrary degree", ACM Trans. Graph., 12, No.1, 1-34 (1993). 18

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