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Given two spheres a spine curve is computed and the rotation minimization blending surface is provided around it.
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Computation of the blending surface of two given spheres is discussed in this paper. The blending surface (or skin), although not uniquely defined in the literature, is normally required to touch the given spheres in plane curves (i.e., in circles). The main advantage of the presented method over the existing ones is the minimization of unwanted di...
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Trapezoidal prestressed unbonded retrofit (TPUR) systems have been recently developed and tested. The authors have already developed a comprehensive and accurate analytical solution for the TPUR system that takes many system parameters into account. The main aim of this paper is to develop a simplified analytical solution for predicting the behavio...
Citations
... Some of them also use evolutionary, iterative approach to find the optimal skin [15,21,22,23], however, these methods are not only time-consuming, but they also depend on the initial position of the skin and may fall into local minima. Other 50 methods use direct real-time computation of the skin, where the central problem is to find the most appropriate touching points on the given circles (in 2D) or touching circles on the given spheres (in 3D) along which the skin will touch the given dataset [1,2,19,24,25]. ...
... As The final parametric surface is rational, since it can use the rational parameterization of the touching circle in its first parameter and will follow the 375 polynomial parameterization of the Hermite arc in its second parameter. Here we note, that the surface can be optimized to avoid unnecessary distorsion by applying rotation-minimizing frames, but it certainly increases the computational time (for details see [24]). ...
Recently there has been a growing interest in the topic of skinning of circles and spheres, since modeling based on these objects has been found useful in areas such as medical applications and character animation. Among others, an efficient method was presented by Kunkli and Hoffmann [1], whilst Bastl et al. also provided an effective skinning algorithm [2]. In this study, we outline the major advantages and disadvantages of these methods, and we show that there are major insufficiencies in [2] in terms of dynamic modeling. We provide a new, improved skinning technique, which preserves the advantages of the two aforementioned algorithms and gives a solution to their arisen problems: it can be used for real-time modeling due to the smooth alteration of skins, and provides good results even in extreme cases. We overcome the problem of self-intersections, and we extend the method to branched systems of circles and spheres.
... [19] and references therein. Due to its technical importance, skinning has attracted the geometric modelling community in recent years and one can find several papers on this topic, see [20,21,22,23]. One of the application areas is computer animation: given a skeletal pose, skinning algorithms are responsible for deforming the geometric skin to respond to the motion of the underlying skeleton. ...
... Consequently, the solution of the linear systems (24) and (25) gives the envelope surface x; see Fig. 7 (bottom, left). Finally, constructing a function f that interpolates (23) and the solution of (24) and (25) and lifting the patch x to R 3,1 , cf. (11), we obtain a parametrization of the RE surface patch y interpolating the points p i and the tangent directions t i1 and t i2 at p i as required; see Fig. 7 (bottom, right). ...
We continue the study of rational envelope (RE) surfaces. Although these surfaces are parametrized with the help of square roots, when considering an RE patch as the medial surface transform in 4D of a spatial domain it yields a rational parametrization of the domain's boundary, i.e., the envelope of the corresponding 2-parameter family of spheres. We formulate efficient algorithms for G1 data interpolation using RE surfaces and apply the developed methods to rational skinning and blending of sets of spheres and cones/cylinders, respectively. Our results are demonstrated on several computed examples of skins and blends with rational parametrizations.
