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The problems of determining the B–spline form of a C2 Pythagorean–hodograph (PH) quintic spline curve interpolating given points, and of using this form to make local modifications, are addressed. To achieve the correct order of continuity, a quintic B–spline basis constructed on a knot sequence in which each (interior) knot is of multiplicity 3 is required. C2 quintic bases on uniform triple knots are constructed for both open and closed C2 curves, and are used to derive simple explicit formulae for the B–spline control points of C2 PH quintic spline curves. These B-spline control points are verified, and generalized to the case of non–uniform knots, by applying a knot removal scheme to the Bézier control points of the individual PH quintic spline segments, associated with a set of six–fold knots. Based on the B–spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed. Only two contiguous spline segments are modified, but to preserve the PH nature of the modified segments, the continuity between modified and unmodified segments must be relaxed from C2 to C1. A number of computed examples are presented, to compare the shape quality of PH quintic and “ordinary” cubic splines subject to control point modifications.

Piecewise polynomial constraint systems are common in numerous problems in computational geometry, such as constraint programming, modeling, and kinematics. We propose a framework that is capable of decomposing, and efficiently solving a wide variety of complex piecewise polynomial constraint systems, that include both zero constraints and inequality constraints, with zero-dimensional or univariate solution spaces. Our framework combines a subdivision-based polynomial solver with a decomposition algorithm in order to handle large and complex systems. We demonstrate the capabilities of our framework on several types of problems and show its performance improvement over a state-of-the-art solver. [ABSTRACT FROM AUTHOR]

With large-scale database systems, statistical analysis of data, occurring in the form of probability distributions, becomes an important task in explorative data analysis. Nevertheless, due to specific properties of density functions, their proper statistical treatment of these data still represents a challenging task in functional data analysis. Namely, the usualmetric does not fully accounts for the relative character of information, carried by density functions; instead, their geometrical features are captured by Bayes spaces of measures. The easiest possibility of expressing density functions in anspace is to use centred logratio transformation, even though this results in functional data with a constant integral constraint that needs to be taken into account in further analysis. While theoretical background for reasonable analysis of density functions is already provided comprehensively by Bayes spaces themselves, preprocessing issues still need to be developed. The aim of this paper is to introduce optimal smoothing splines for centred logratio transformed density functions that take all their specific features into account and provide a concise methodology for reasonable preprocessing of raw (discretized) distributional observations. Theoretical developments are illustrated with a real-world data set from official statistics and with a simulation study. [ABSTRACT FROM PUBLISHER]

Abstract: We present a method for constructing blades of hydroelectric turbines and ship propellers based on design parameters that possess a clear hydraulic meaning. The design process corresponds to the classical construction of a blade using the medial surface of the blade and profile curves attached to it. The main new contribution of the paper consists in realizing this construction using B-spline techniques. In particular, it is shown how to obtain blade boundary surfaces (which describe the pressure and the suction side of the blade) which are joined with C 1-smoothness along the leading edge. Moreover, special attention is paid to the construction of propeller blades with a well-defined tangent plane at the tip, which is a singular point of the blade boundary surfaces. In order to guarantee these smoothness properties, we generate and analyze singularly parameterized medial surfaces. We contribute novel shape modeling techniques that are based on singular parameterizations and demonstrate their potential for applications in industry. Finally, it is shown how to represent the blades as B-spline surfaces with a relatively small number of control points. [Copyright &y& Elsevier]

This paper is to discuss an approach which combines B-spline patches and transfinite interpolation to establish a linear algebraic system for solving partial differential equations and modify the WEB-spline method developed by Klaus Hollig to derive this new idea. First of all, the authors replace the R-function method with transfinite interpolation to build a function which vanishes on boundaries. Secondly, the authors simulate the partial differential equation by directly applying differential operators to basis functions, which is similar to the RBF method rather than Hollig's method. These new strategies then make the constructing of bases and the linear system much more straightforward. And as the interpolation is brought in, the design of schemes for solving practical PDEs can be more flexible. This new method is easy to carry out and suitable for simulations in the fields such as graphics to achieve rapid rendering. Especially when the specified precision is not very high, this method performs much faster than WEB-spline method. [ABSTRACT FROM AUTHOR]

In this work we propose the use of B-spline functions for the parametric representation of high resolution images from low sampled data in the Fourier domain. Traditionally, exponential basis functions are employed in this situation, but they produce artifacts and amplify the noise on the data. We present the method in an algorithmic form and carefully consider the problem of solving the ill-conditioned linear system arising from the method by an efficient regularization method. Two applications of the proposed method to dynamic Magnetic Resonance images are considered. Dynamic Magnetic Resonance acquires a time series of images of the same slice of the body; in order to fasten the acquisition, the data are low sampled in the Fourier space. Numerical experiments have been performed both on simulated and real Magnetic Resonance data. They show that the B-splines reduce the artifacts and the noise in the representation of high resolution Magnetic Resonance images from low sampled data. [ABSTRACT FROM AUTHOR]