In applications that involve interactive curve and surface modeling, the intuitive manipulation of shapes is crucial. For instance, user interaction is facilitated if a geometrical object can be manipulated through control points that interpolate the shape itself. Additionally, models for shape representation often need to provide local shape control and they need to be able to reproduce common shape primitives such as ellipsoids, spheres, cylinders, or tori. We present a general framework to construct families of compactly-supported interpolators that are piecewise-exponential polynomial. They can be designed to satisfy regularity constraints of any order and they enable one to build parametric deformable shape models by suitable linear combinations of interpolators. They allow to change the resolution of shapes based on the refinability of B-splines. We illustrate their use on examples to construct shape models that involve curves and surfaces with applications to interactive modeling and character design.
Recently, a method to build piecewise-polynomial interpolators has been presented in @cite_23 @cite_19 and its bivariate generalization was proposed in @cite_12 . The present work is the continuation of our previous efforts to, first, generalize the popular Catmull-Rom @cite_15 and Keys @cite_24 @cite_33 interpolators for practical applications @cite_37 @cite_2 @cite_4 @cite_36 and, next, to go one step further and construct families of interpolators that allow varying the resolution of a shape @cite_22 @cite_13 . Here, the novelty w.r.t. @cite_36 is that the presented framework allows one to vary the resolution of shapes which facilitates shape design in practice, as illustrated in .
@cite_37 @cite_4 @cite_33 @cite_22 @cite_36 @cite_24 @cite_19 @cite_23 @cite_2 @cite_15 @cite_13 @cite_12
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In this paper we consider the problem of designing piecewise polynomial local interpolants of non-uniformly spaced data. We provide a constructive approach that, for any assigned degree of polynomial reproduction, continuity order, and support width, allows for generating the fundamental spline functions of minimum degree having the desired properties. Finally, the proposed construction is extended to handle open sets of data and to the case of multiple knots. This paper demonstrates that the computational effort required to develop numerical solutions to continuous-state dynamic programs can be reduced significantly when cubic piecewise polynomial functions, rather than tensor product linear interpolants, are used to approximate the value function. Tensor product cubic splines, represented in either piecewise polynomial or B-spline form, and multivariate Hermite polynomials are considered. Computational savings are possible because of the improved accuracy of higher-order functions and because the smoothness of higher-order functions allows efficient quasi-Newton methods to be used to compute optimal decisions. The use of the more efficient piecewise polynomial form of the spline was slightly superior to the use of Hermite polynomials for the test problem and easier to program. In comparison to linear interpolation, use of splines in piecewise polynomial form reduced the CPU time to obtain results of equivalent accuracy by a factor of 250–330 for a stochastic 4-dimensional water supply reservoir problem with a smooth objective function, and factors ranging from 25–400 for a sequence of 2-, 3-, 4-, and 5-dimensional problems. As a result, a problem that required two hours to solve with linear interpolation was solved in a less than a minute with spline interpolation with no loss of accuracy. In CAGD the design of a surface that interpolates an arbitrary quadrilateral mesh is definitely a challenging task. The basic requirement is to satisfy both criteria concerning the regularity of the surface and aesthetic concepts.With regard to aesthetic quality, it is well known that interpolatory methods often produce shape artifacts when the data points are unevenly spaced. In the univariate setting, this problem can be overcome, or at least mitigated, by exploiting a proper non-uniform parametrization, that accounts for the geometry of the data. Recently the same principle has been generalized and proven to be effective in the context of bivariate interpolatory subdivision schemes.In this paper, we propose a construction for parametric surfaces of good aesthetic quality and high smoothness interpolating quadrilateral meshes of arbitrary topology. In the classical tensor product setting the same parameter interval must be shared by an entire row or column of mesh edges. Conversely, in this paper, we assign a different parameter interval to each mesh edge. This particular structure, which we call an augmented parametrization, allows us to interpolate each section polyline at parameters values that prevent wiggling of the resulting curve or other interpolation artifacts and yields high quality interpolatory surfaces.The proposed method is generalization of the local univariate spline interpolants introduced in (2013a) and (2014), that have arbitrary continuity and arbitrary order of polynomial reproduction. The generated surfaces retain the same smoothness of the underlying class of univariate splines in the regular regions of the mesh (where, locally, all vertices have valence 4). Mesh regions containing vertices of valence other than 4 are covered with suitably defined surface patches joining the neighboring regular ones with G 1 - or G 2 -continuity. Interpolatory methods often produce shape artifacts for unevenly-spaced data points.A proper non-uniform parametrization can help overcome interpolation artifacts.Parametric surfaces of good aesthetic quality and high smoothness are constructed.The method relies on non-uniform parametrization and locally supported splines.The surfaces interpolate quadrilateral meshes with arbitrary topology. This paper presents a general framework for the construction of piecewise-polynomial local interpolants with given smoothness and approximation order, defined on non-uniform knot partitions. We design such splines through a suitable combination of polynomial interpolants with either polynomial or rational, compactly supported blending functions. In particular, when the blending functions are rational, our approach provides spline interpolants having low, and sometimes minimum degree. Thanks to its generality, the proposed framework also allows us to recover uniform local interpolating splines previously proposed in the literature, to generalize them to the non-uniform case, and to complete families of arbitrary support width. Furthermore it provides new local interpolating polynomial splines with prescribed smoothness and polynomial reproduction properties.