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S0165168415000067 | Sketch-based human motion retrieval is a hot topic in computer animation in recent years. In this paper, we present a novel sketch-based human motion retrieval method via selected 2-dimensional (2D) Geometric Posture Descriptor (2GPD). Specially, we firstly propose a rich 2D pose feature call 2D Geometric Posture Descriptor (2GPD), which is effective in encoding the 2D posture similarity by exploiting the geometric relationships among different human body parts. Since the original 2GPD is of high dimension and redundant, a semi-supervised feature selection algorithm derived from Laplacian Score is then adopted to select the most discriminative feature component of 2GPD as feature representation, and we call it as selected 2GPD. Finally, a posture-by-posture motion retrieval algorithm is used to retrieve a motion sequence by sketching several key postures. Experimental results on CMU human motion database demonstrate the effectiveness of our proposed approach. | Sketch-based human motion retrieval via selected 2D geometric posture descriptor |
S0165168415001073 | In this paper, we address the tasks of audio source counting and separation for a stereo anechoic mixture of audio signals. This will be achieved in two stages. In the first stage, a novel approach is introduced for estimating the number of sources as well as the channel mixing coefficients. For this purpose, a 2-D spectrum is evaluated against both the phase and amplitude differences of the two channels. Hence, obtaining the peak locations of the spectrum yields the number of the sources and the corresponding channel coefficients. In the second stage, an extension of a single channel complex matrix factorization method to multichannel is developed to extract the individual source signals. We find primary estimates of the sources via binary masking and then apply the complex factorization to the complex spectrogram of each source. The obtained factors are then utilized as initial values in the complex multichannel factorization model. We also suggest a method for estimating the number of required components for modeling each source. The separation performance improvement over the conventional methods is investigated by calculating BSS evaluation metrics. The comparison is also carried out in terms of source counting and localization with the recently proposed DeMIX-Anechoic method. | Blind audio source counting and separation of anechoic mixtures using the multichannel complex NMF framework |
S0165168415003187 | The fractional Fourier transform (FRFT) is one of the most useful tools for the nonstationary signal processing. In this paper, the randomized nonuniform sampling and approximate reconstruction of the nonstationary random signals in the fractional Fourier domain (FRFD) are developed. The nonuniform samples are treated as random perturbations from a uniform grid. The samples used for the sinc interpolation reconstruction are placed on another nonuniform grid which is not necessarily equal to the samples originally acquired. When considering the second-order random statistic characters, the nonuniform sampling is equivalent to the uniform sampling of the signal after a pre-filter in the FRFD, where the frequency response is related to the characteristic function (with its argument scaled by csc α ) of the perturbations. The effectiveness of the reconstruction is analyzed and the mean square error (MSE) is computed by utilizing the equivalent filter system. Furthermore, the randomized reconstruction of the chirp period stationary random signal is proposed. At last, the minimum MSE on the special cases of the randomized sampling and reconstruction is discussed. The effectiveness of the proposed reconstruction method is verified by the simulation. | Randomized nonuniform sampling and reconstruction in fractional Fourier domain |
S0165168416000451 | In signal processing applications, it is often necessary to extract oscillatory components and their properties from time–frequency representations, e.g. the windowed Fourier transform or wavelet transform. The first step in this procedure is to find an appropriate ridge curve: a sequence of amplitude peak positions (ridge points), corresponding to the component of interest and providing a measure of its instantaneous frequency. This is not a trivial issue, and the optimal method for extraction is still not settled or agreed. We discuss and develop procedures that can be used for this task and compare their performance on both simulated and real data. In particular, we propose a method which, in contrast to many other approaches, is highly adaptive so that it does not need any parameter adjustment for the signal to be analyzed. Being based on dynamic path optimization and fixed point iteration, the method is very fast, and its superior accuracy is also demonstrated. In addition, we investigate the advantages and drawbacks that synchrosqueezing offers in relation to curve extraction. The codes used in this work are freely available for download. | Extraction of instantaneous frequencies from ridges in time–frequency representations of signals |
S0167639314000156 | The RSR2015 database, designed to evaluate text-dependent speaker verification systems under different durations and lexical constraints has been collected and released by the Human Language Technology (HLT) department at Institute for Infocomm Research (I2R) in Singapore. English speakers were recorded with a balanced diversity of accents commonly found in Singapore. More than 151h of speech data were recorded using mobile devices. The pool of speakers consists of 300 participants (143 female and 157 male speakers) between 17 and 42years old making the RSR2015 database one of the largest publicly available database targeted for text-dependent speaker verification. We provide evaluation protocol for each of the three parts of the database, together with the results of two speaker verification system: the HiLAM system, based on a three layer acoustic architecture, and an i-vector/PLDA system. We thus provide a reference evaluation scheme and a reference performance on RSR2015 database to the research community. The HiLAM outperforms the state-of-the-art i-vector system in most of the scenarios. | Text-dependent speaker verification: Classifiers, databases and RSR2015 |
S0167639314000697 | A spectral envelope estimation algorithm is presented to achieve high-quality speech synthesis. The concept of the algorithm is to obtain an accurate and temporally stable spectral envelope. The algorithm uses fundamental frequency (F0) and consists of F0-adaptive windowing, smoothing of the power spectrum, and spectral recovery in the quefrency domain. Objective and subjective evaluations were carried out to demonstrate the effectiveness of the proposed algorithm. Results of both evaluations indicated that the proposed algorithm can obtain a temporally stable spectral envelope and synthesize speech with higher sound quality than speech synthesized with other algorithms. | CheapTrick, a spectral envelope estimator for high-quality speech synthesis |
S0167639314000740 | With teleconferencing becoming more accessible as a communication platform, researchers are working to understand the consequences of the interaction between human perception and this unfamiliar environment. Given the enclosed space of a teleconference room, along with the physical separation between the user, microphone and speakers, the transmitted audio often becomes mixed with the reverberating auditory components from the room. As a result, the audio can be perceived as smeared in time, and this can affect the user experience and perceived quality. Moreover, other challenges remain to be solved. For instance, during encoding, compression and transmission, the audio and video streams are typically treated separately. Consequently, the signals are rarely perfectly aligned and synchronous. In effect, timing affects both reverberation and audiovisual synchrony, and the two challenges may well be inter-dependent. This study explores the temporal integration of audiovisual continuous speech and speech syllables, along with a non-speech event, across a range of asynchrony levels for different reverberation conditions. Non-reverberant stimuli are compared to stimuli with added reverberation recordings. Findings reveal that reverberation does not affect the temporal integration of continuous speech. However, reverberation influences the temporal integration of the isolated speech syllables and the action-oriented event, with perceived subjective synchrony skewed towards audio lead asynchrony and away from the more common audio lag direction. Furthermore, less time is spent on simultaneity judgements for the longer sequences when the temporal offsets get longer and when reverberation is introduced, suggesting that both asynchrony and reverberation add to the demands of the task. | Audiovisual temporal integration in reverberant environments |
S0167639315000692 | This paper presents an unsupervised method that allows for gradual interpolation between language varieties in statistical parametric speech synthesis using Hidden Semi-Markov Models (HSMMs). We apply dynamic time warping using Kullback–Leibler divergence on two sequences of HSMM states to find adequate interpolation partners. The method operates on state sequences with explicit durations and also on expanded state sequences where each state corresponds to one feature frame. In an intelligibility and dialect rating subjective evaluation of synthesized test sentences, we show that our method can generate intermediate varieties for three Austrian dialects (Viennese, Innervillgraten, Bad Goisern). We also provide an extensive phonetic analysis of the interpolated samples. The analysis includes input-switch rules, which cover historically different phonological developments of the dialects versus the standard language; and phonological processes, which are phonetically motivated, gradual, and common to all varieties. We present an extended method which linearly interpolates phonological processes but uses a step function for input-switch rules. Our evaluation shows that the integration of this kind of phonological knowledge improves dialect authenticity judgment of the synthesized speech, as performed by dialect speakers. Since gradual transitions between varieties are an existing phenomenon, we can use our methods to adapt speech output systems accordingly. | Unsupervised and phonologically controlled interpolation of Austrian German language varieties for speech synthesis |
S0167642313000452 | The CancerGrid approach to software support for clinical trials is based on two principles: careful curation of semantic metadata about clinical observations, to enable subsequent data integration, and model-driven generation of trial-specific software artefacts from a trial protocol, to streamline the software development process. This paper explains the approach, presents four varied case studies, and discusses the lessons learned. | The CancerGrid experience: Metadata-based model-driven engineering for clinical trials |
S0167642314000112 | We compare different algorithms for computing eigenvalues and eigenvectors of a symmetric band matrix across a wide range of synthetic test problems. Of particular interest is a comparison of state-of-the-art tridiagonalization-based methods as implemented in Lapack or Plasma on the one hand, and the block divide-and-conquer (BD&C) algorithm as well as the block twisted factorization (BTF) method on the other hand. The BD&C algorithm does not require tridiagonalization of the original band matrix at all, and the current version of the BTF method tridiagonalizes the original band matrix only for computing the eigenvalues. Avoiding the tridiagonalization process sidesteps the cost of backtransformation of the eigenvectors. Beyond that, we discovered another disadvantage of the backtransformation process for band matrices: In several scenarios, a lot of gradual underflow is observed in the (optional) accumulation of the transformation matrix and in the (obligatory) backtransformation step. According to the IEEE 754 standard for floating-point arithmetic, this implies many operations with subnormal (denormalized) numbers, which causes severe slowdowns compared to the other algorithms without backtransformation of the eigenvectors. We illustrate that in these cases the performance of existing methods from Lapack and Plasma reaches a competitive level only if subnormal numbers are disabled (and thus the IEEE standard is violated). Overall, our performance studies illustrate that if the problem size is large enough relative to the bandwidth, BD&C tends to achieve the highest performance of all methods if the spectrum to be computed is clustered. For test problems with well separated eigenvalues, the BTF method tends to become the fastest algorithm with growing problem size. | Comparison of eigensolvers for symmetric band matrices |
S0167642315000659 | A hierarchical approach for modelling the adaptability features of complex systems is introduced. It is based on a structural level S, describing the adaptation dynamics of the system, and a behavioural level B accounting for the description of the admissible dynamics of the system. Moreover, a unified system, called S [ B ] , is defined by coupling S and B. The adaptation semantics is such that the S level imposes structural constraints on the B level, which has to adapt whenever it no longer can satisfy them. In this context, we introduce weak and strong adaptability, i.e. the ability of a system to adapt for some evolution paths or for all possible evolutions, respectively. We provide a relational characterisation for these two notions and we show that adaptability checking, i.e. deciding if a system is weakly or strongly adaptable, can be reduced to a CTL model checking problem. We apply the model and the theoretical results to the case study of a motion controller of autonomous transport vehicles. | Adaptability checking in complex systems |
S0167642315001288 | In order to provide a rigorous foundation for Software Product Lines (SPLs), several fundamental approaches have been proposed to their formal behavioral modeling. In this paper, we provide a structured overview of those formalisms based on labeled transition systems and compare their expressiveness in terms of the set of products they can specify. Moreover, we define the notion of tests for each of these formalisms and show that our notions of testing precisely capture product derivation, i.e., all valid products will pass the set of test cases of the product line and each invalid product fails at least one test case of the product line. | Basic behavioral models for software product lines: Expressiveness and testing pre-orders |
S0167819113001051 | SpiNNaker is a biologically-inspired massively-parallel computer designed to model up to a billion spiking neurons in real-time. A full-fledged implementation of a SpiNNaker system will comprise more than 105 integrated circuits (half of which are SDRAMs and half multi-core systems-on-chip). Given this scale, it is unavoidable that some components fail and, in consequence, fault-tolerance is a foundation of the system design. Although the target application can tolerate a certain, low level of failures, important efforts have been devoted to incorporate different techniques for fault tolerance. This paper is devoted to discussing how hardware and software mechanisms collaborate to make SpiNNaker operate properly even in the very likely scenario of component failures and how it can tolerate system-degradation levels well above those expected. | SpiNNaker: Fault tolerance in a power- and area- constrained large-scale neuromimetic architecture |
S0167819114000337 | Linear least squares problems are commonly solved by QR factorization. When multiple solutions need to be computed with only minor changes in the underlying data, knowledge of the difference between the old data set and the new can be used to update an existing factorization at reduced computational cost. We investigate the viability of implementing QR updating algorithms on GPUs and demonstrate that GPU-based updating for removing columns achieves speed-ups of up to 13.5× compared with full GPU QR factorization. We characterize the conditions under which other types of updates also achieve speed-ups. | Implementing QR factorization updating algorithms on GPUs |
S0167819114000398 | Simulation of in vivo cellular processes with the reaction–diffusion master equation (RDME) is a computationally expensive task. Our previous software enabled simulation of inhomogeneous biochemical systems for small bacteria over long time scales using the MPD-RDME method on a single GPU. Simulations of larger eukaryotic systems exceed the on-board memory capacity of individual GPUs, and long time simulations of modest-sized cells such as yeast are impractical on a single GPU. We present a new multi-GPU parallel implementation of the MPD-RDME method based on a spatial decomposition approach that supports dynamic load balancing for workstations containing GPUs of varying performance and memory capacity. We take advantage of high-performance features of CUDA for peer-to-peer GPU memory transfers and evaluate the performance of our algorithms on state-of-the-art GPU devices. We present parallel efficiency and performance results for simulations using multiple GPUs as system size, particle counts, and number of reactions grow. We also demonstrate multi-GPU performance in simulations of the Min protein system in E. coli. Moreover, our multi-GPU decomposition and load balancing approach can be generalized to other lattice-based problems. | Simulation of reaction diffusion processes over biologically relevant size and time scales using multi-GPU workstations |
S0167819114000659 | In this paper, we develop a rank-mapping algorithm for an icosahedral grid system on a massive parallel computer with the 3-D torus network topology, specifically on the K computer. Our aim is to improve the weak scaling performance of the point-to-point communications for exchanging grid-point values between adjacent grid regions on a sphere. We formulate a new rank-mapping algorithm to reduce the maximum number of hops for the point-to-point communications. We evaluate both the new algorithm and the standard ones on the K computer, using the communication kernel of the Nonhydrostatic Icosahedral Atmospheric Model (NICAM), a global atmospheric model with an icosahedral grid system. We confirm that, unlike the standard algorithms, the new one achieves almost perfect performance in the weak scaling on the K computer, even for 10,240 nodes. Results of additional experiments imply that the high scalability of the new rank-mapping algorithm on the K computer is achieved by reducing network congestion in the links between adjacent nodes. | Scalable rank-mapping algorithm for an icosahedral grid system on the massive parallel computer with a 3-D torus network |
S0167819114000842 | Data scientists have applied various analytic models and techniques to address the oft-cited problems of large volume, high velocity data rates and diversity in semantics. Such approaches have traditionally employed analytic techniques in a streaming or batch processing paradigm. This paper presents CRUCIBLE, a first-in-class framework for the analysis of large-scale datasets that exploits both streaming and batch paradigms in a unified manner. The CRUCIBLE framework includes a domain specific language for describing analyses as a set of communicating sequential processes, a common runtime model for analytic execution in multiple streamed and batch environments, and an approach to automating the management of cell-level security labelling that is applied uniformly across runtimes. This paper shows the applicability of CRUCIBLE to a variety of state-of-the-art analytic environments, and compares a range of runtime models for their scalability and performance against a series of native implementations. The work demonstrates the significant impact of runtime model selection, including improvements of between 2.3× and 480× between runtime models, with an average performance gap of just 14× between CRUCIBLE and a suite of equivalent native implementations. | Towards unified secure on- and off-line analytics at scale |
S0167819114001148 | We introduce a region template abstraction and framework for the efficient storage, management and processing of common data types in analysis of large datasets of high resolution images on clusters of hybrid computing nodes. The region template abstraction provides a generic container template for common data structures, such as points, arrays, regions, and object sets, within a spatial and temporal bounding box. It allows for different data management strategies and I/O implementations, while providing a homogeneous, unified interface to applications for data storage and retrieval. A region template application is represented as a hierarchical dataflow in which each computing stage may be represented as another dataflow of finer-grain tasks. The execution of the application is coordinated by a runtime system that implements optimizations for hybrid machines, including performance-aware scheduling for maximizing the utilization of computing devices and techniques to reduce the impact of data transfers between CPUs and GPUs. An experimental evaluation on a state-of-the-art hybrid cluster using a microscopy imaging application shows that the abstraction adds negligible overhead (about 3%) and achieves good scalability and high data transfer rates. Optimizations in a high speed disk based storage implementation of the abstraction to support asynchronous data transfers and computation result in an application performance gain of about 1.13 × . Finally, a processing rate of 11,730 4K × 4K tiles per minute was achieved for the microscopy imaging application on a cluster with 100 nodes (300 GPUs and 1200 CPU cores). This computation rate enables studies with very large datasets. | Region templates: Data representation and management for high-throughput image analysis |
S0167819115000095 | The human brain is a complex biological neural network characterised by high degrees of connectivity among neurons. Any system designed to simulate large-scale spiking neuronal networks needs to support such connectivity and the associated communication traffic in the form of spike events. This paper investigates how best to generate multicast routes for SpiNNaker, a purpose-built, low-power, massively-parallel architecture. The discussed algorithms are an essential ingredient for the efficient operation of SpiNNaker since generating multicast routes is known to be an NP-complete problem. In fact, multicast communications have been extensively studied in the literature, but we found no existing algorithm adaptable to SpiNNaker. The proposed algorithms exploit the regularity of the two-dimensional triangular torus topology and the availability of selective multicast at hardware level. A comprehensive study of the parameters of the algorithms and their effectiveness is carried out in this paper considering different destination distributions ranging from worst-case to a real neural application. The results show that two novel proposed algorithms can reduce significantly the pressure exerted onto the interconnection infrastructure while remaining effective to be used in a production environment. | SpiNNaker: Enhanced multicast routing |
S0167819115000472 | Community detection has become a fundamental operation in numerous graph-theoretic applications. It is used to reveal natural divisions that exist within real world networks without imposing prior size or cardinality constraints on the set of communities. Despite its potential for application, there is only limited support for community detection on large-scale parallel computers, largely owing to the irregular and inherently sequential nature of the underlying heuristics. In this paper, we present parallelization heuristics for fast community detection using the Louvain method as the serial template. The Louvain method is a multi-phase, iterative heuristic for modularity optimization. Originally developed by Blondel et al. (2008), the method has become increasingly popular owing to its ability to detect high modularity community partitions in a fast and memory-efficient manner. However, the method is also inherently sequential, thereby limiting its scalability. Here, we observe certain key properties of this method that present challenges for its parallelization, and consequently propose heuristics that are designed to break the sequential barrier. For evaluation purposes, we implemented our heuristics using OpenMP multithreading, and tested them over real world graphs derived from multiple application domains (e.g., internet, citation, biological). Compared to the serial Louvain implementation, our parallel implementation is able to produce community outputs with a higher modularity for most of the inputs tested, in comparable number or fewer iterations, while providing absolute speedups of up to 16 × using 32 threads. | Parallel heuristics for scalable community detection |
S0167839613000356 | Motivated by applications in freeform architecture, we study surfaces which are composed of smoothly joined bilinear patches. These surfaces turn out to be discrete versions of negatively curved affine minimal surfaces and share many properties with their classical smooth counterparts. We present computational design approaches and study special cases which should be interesting for the architectural application. | Smooth surfaces from bilinear patches: Discrete affine minimal surfaces |
S0167839613000368 | A characterization for spatial Pythagorean-hodograph (PH) curves of degree 7 with rotation-minimizing EulerâRodrigues frames (ERFs) is determined, in terms of one real and two complex constraints on the curve coefficients. These curves can interpolate initial/final positions p i and p f and orientational frames ( t i , u i , v i ) and ( t f , u f , v f ) so as to define a rational rotation-minimizing rigid body motion. Two residual free parameters, that determine the magnitudes of the end derivatives, are available for optimizing shape properties of the interpolant. This improves upon existing algorithms for quintic PH curves with rational rotation-minimizing frames (RRMF quintics), which offer no residual freedoms. Moreover, the degree 7 PH curves with rotation-minimizing ERFs are capable of interpolating motion data for which the RRMF quintics do not admit real solutions. Although these interpolants are of higher degree than the RRMF quintics, their rotation-minimizing frames are actually of lower degree (6 versus 8), since they coincide with the ERF. This novel construction of rational rotation-minimizing motions may prove useful in applications such as computer animation, geometric sweep operations, and robot trajectory planning. | Rotation-minimizing Euler-Rodrigues rigid-body motion interpolants |
S0167839613000502 | We present a fast algorithm for finding a μ-basis for any rational planar curve that has a complex rational parametrization. We begin by identifying two canonical syzygies that can be extracted directly from any complex rational parametrization without performing any additional calculations. For generic complex rational parametrizations, these two special syzygies form a μ-basis for the corresponding real rational curve. In those anomalous cases where these two canonical syzygies do not form a μ-basis, we show how to quickly calculate a μ-basis by performing Gaussian elimination on these two special syzygies. We also present an algorithm to determine if a real rational planar curve has a complex rational parametrization. Examples are provided to illustrate our methods. | μ-Bases for complex rational curves |
S0167839613000514 | This paper is devoted to introducing a new approach for computing the topology of a real algebraic plane curve presented either parametrically or defined by its implicit equation when the corresponding polynomials which describe the curve are known only “by values”. This approach is based on the replacement of the usual algebraic manipulation of the polynomials (and their roots) appearing in the topology determination of the given curve with the computation of numerical matrices (and their eigenvalues). Such numerical matrices arise from a typical construction in Elimination Theory known as the Bézout matrix which in our case is specified by the values of the defining polynomial equations on several sample points. | Computing the topology of a real algebraic plane curve whose defining equations are available only “by values” |
S0167839613000538 | Spiral segments are useful in the design of fair curves. They are important in CAD/CAM applications, the design of highway and railway routes, trajectories of mobile robots and other similar applications. The quintic Pythagorean-hodograph (PH) curve discussed in this article is polynomial; it has the attractive properties that its arc-length is a polynomial of its parameter, and the formula for its offset is a rational algebraic expression. This paper generalises earlier results on planar PH quintic spiral segments and examines techniques for designing fair curves using the new results. | Curve design with more general planar Pythagorean-hodograph quintic spiral segments |
S0167839613000551 | This paper proposes a generalization of the ordinary de Casteljau algorithm to manifold-valued data including an important special case which uses the exponential map of a symmetric space or Riemannian manifold. We investigate some basic properties of the corresponding Bézier curves and present applications to curve design on polyhedra and implicit surfaces as well as motion of rigid body and positive definite matrices. Moreover, we apply our approach to construct canal and developable surfaces. | De Casteljauʼs algorithm on manifolds |
S0167839613000575 | Recently a new approach to piecewise polynomial spaces generated by B-spline has been presented by T. Dokken, T. Lyche and H.F. Pettersen, namely Locally Refined splines. In their recent work (Dokken et al., 2013) they define the LR B-spline collection and provide tools to compute the space dimension. Here different properties of the LR-splines are analyzed: in particular the coefficients for polynomial representations and their relation with other properties such as linear independence and the number of B-splines covering each element. | Some properties of LR-splines |
S0167839613000587 | We examine the problem of computing exactly the Voronoi diagram (via the dual Delaunay graph) of a set of, possibly intersecting, smooth convex pseudo-circles in the Euclidean plane, given in parametric form. Pseudo-circles are (convex) sites, every pair of which has at most two intersecting points. The Voronoi diagram is constructed incrementally. Our first contribution is to propose robust and efficient algorithms, under the exact computation paradigm, for all required predicates, thus generalizing earlier algorithms for non-intersecting ellipses. Second, we focus on InCircle, which is the hardest predicate, and express it by a simple sparse 5 × 5 polynomial system, which allows for an efficient implementation by means of successive Sylvester resultants and a new factorization lemma. The third contribution is our cgal-based c++ software for the case of possibly intersecting ellipses, which is the first exact implementation for the problem. Our code spends about a minute to construct the Voronoi diagram of 200 ellipses, when few degeneracies occur. It is faster than the cgal segment Voronoi diagram, when ellipses are approximated by k-gons for k > 15 , and a state-of-the-art implementation of the Voronoi diagram of points, when each ellipse is approximated by more than 1250 points. | Exact Voronoi diagram of smooth convex pseudo-circles: General predicates, and implementation for ellipses |
S0167839613000678 | In this paper, a new method for computing intersection between a ray and a parametric surface is proposed, which finds many applications in computer graphics, robotics and geometric modeling. The method uses the second order derivative of the surface, which can handle inherent problems that Newton–Raphson and Halley methods have such as instability caused by inappropriate initial conditions and tangential intersection. Case examples are presented to demonstrate the capability of the proposed method. | A second order geometric method for ray/parametric surface intersection |
S0167839613000782 | T-splines are a generalization of NURBS surfaces, the control meshes of which allow T-junctions. T-splines can significantly reduce the number of superfluous control points in NURBS surfaces, and provide valuable operations such as local refinement and merging of several B-splines surfaces in a consistent framework. In this paper, we propose a variant of T-splines called Modified T-splines. The basic idea is to construct a set of basis functions for a given T-mesh that have the following nice properties: non-negativity, linear independence, partition of unity and compact support. Due to the good properties of the basis functions, the Modified T-splines are favorable both in adaptive geometric modeling and isogeometric analysis. | Modified T-splines |
S0167839613000794 | A method to construct arbitrary order continuous curves, which pass through a given set of data points, is introduced. The method can derive a new family of symmetric interpolating splines with various nice properties, such as partition of unity, interpolation property, local support and second order precision etc. Applying these new splines to construct curves and surfaces, one can adjust the shape of the constructed curve and surface locally by moving some interpolating points or by inserting new interpolating points. Constructing closed smooth curves and surfaces and smooth joining curves and surfaces also become very simple, in particular, for constructing C r ( r ⩾ 1 ) continuous closed surfaces by using the repeating technique. These operations mentioned do not require one to solve a system of equations. The resulting curves or surfaces are directly expressed by the basis spline functions. Furthermore, the method can also directly produce control points of the interpolating piecewise Bézier curves or tensor product Bézier surfaces by using matrix formulas. Some examples are given to support the conclusions. | Uniform interpolation curves and surfaces based on a family of symmetric splines |
S0167839613000800 | Smooth freeform skins from simple panels constitute a challenging topic arising in contemporary architecture. We contribute to this problem area by showing how to approximate a negatively curved surface by smoothly joined rational bilinear patches. The approximation problem is solved with help of a new computational approach to the hyperbolic nets of Huhnen-Venedey and Rörig and optimization algorithms based on it. We also discuss its limits which lie in the topology of the input surface. Finally, freeform deformations based on Darboux transformations are used to generate smooth surfaces from smoothly joined Darboux cyclide patches; in this way we eliminate the restriction to surfaces with negative Gaussian curvature. | Smooth surfaces from rational bilinear patches |
S0167839613001003 | An orthonormal frame ( f 1 , f 2 , f 3 ) is rotation-minimizing with respect to f i if its angular velocity ω satisfies ω ⋅ f i ≡ 0 — or, equivalently, the derivatives of f j and f k are both parallel to f i . The Frenet frame ( t , p , b ) along a space curve is rotation-minimizing with respect to the principal normal p, and in recent years adapted frames that are rotation-minimizing with respect to the tangent t have attracted much interest. This study is concerned with rotation-minimizing osculating frames ( f , g , b ) incorporating the binormal b, and osculating-plane vectors f, g that have no rotation about b. These frame vectors may be defined through a rotation of t, p by an angle equal to minus the integral of curvature with respect to arc length. In aeronautical terms, the rotation-minimizing osculating frame (RMOF) specifies yaw-free rigid-body motion along a curved path. For polynomial space curves possessing rational Frenet frames, the existence of rational RMOFs is investigated, and it is found that they must be of degree 7 at least. The RMOF is also employed to construct a novel type of ruled surface, with the property that its tangent planes coincide with the osculating planes of a given space curve, and its rulings exhibit the least possible rate of rotation consistent with this constraint. | Rotation-minimizing osculating frames |
S0167839613001015 | A construction of spline spaces suitable for representing smooth parametric surfaces of arbitrary topological genus and arbitrary order of continuity is proposed. The obtained splines are a direct generalization of bivariate polynomial splines on planar partitions. They are defined as composite functions consisting of rational functions and are parametrized by a single parameter domain, which is a piecewise planar surface, such as a triangulation of a cloud of 3D points. The idea of the construction is to utilize linear rational transformations (or transition maps) to endow the piecewise planar surface with a particular C ∞ -differentiable structure appropriate for defining rational splines. | RAGS: Rational geometric splines for surfaces of arbitrary topology |
S0167839613001027 | In this paper, the dual representation of spatial parametric curves and its properties are studied. In particular, rational curves have a polynomial dual representation, which turns out to be both theoretically and computationally appropriate to tackle the main goal of the paper: spatial rational Pythagorean-hodograph curves (PH curves). The dual representation of a rational PH curve is generated here by a quaternion polynomial which defines the Euler–Rodrigues frame of a curve. Conditions which imply low degree dual form representation are considered in detail. In particular, a linear quaternion polynomial leads to cubic or reparameterized cubic polynomial PH curves. A quadratic quaternion polynomial generates a wider class of rational PH curves, and perhaps the most useful is the ten-parameter family of cubic rational PH curves, determined here in the closed form. | Dual representation of spatial rational Pythagorean-hodograph curves |
S0167839613001039 | In this work, an extension has been performed on the analysis basis of spline-based meshfree method (SBMFM) to stabilize its solution. The potential weakness of the SBMFM is its numerical instability from using regular grid background mesh. That is, if an extremely small trimmed element is produced by the trimming curves that represent boundaries of the analysis domain, it can induce an excessively large condition number in global system matrix. To resolve the instability problem, the extension technique of the weighted extended B-spline (WEB-spline) is implemented in the SBMFM. The basis functions with very small trimmed supports are extrapolated by neighboring basis functions with some special scheme so that those basis functions can be condensed in the solution process. In order to impose essential boundary conditions in the SBMFM with extended basis, Nitsche's method is implemented. Using numerical examples, the presented SBMFM with extended basis is shown to be valid and effective. Moreover, the condition number of the system is well-managed guaranteeing the stability of the numerical analysis. | Spline-based meshfree method with extended basis |
S0167839613001040 | A new binary four-point subdivision scheme is presented, which keeps the second-order divided difference at the old vertices unchanged when the new vertices are inserted. Using the symbol of the subdivision scheme, we show that the limit curve is at least C 3 continuous. Furthermore, the conditions imposed on the initial points are also discussed, thus the given limit functions are both monotonicity preserving and convexity preserving | A new four-point shape-preserving C 3 subdivision scheme |
S0167839613001052 | We develop a method for computing all the generalized asymptotes of a real plane algebraic curve C implicitly defined by an irreducible polynomial f ( x , y ) ∈ R [ x , y ] . The approach is based on the notion of perfect curve introduced from the concepts and results presented in Blasco and Pérez-Díaz (2013). In addition, we study some properties concerning perfect curves and in particular, we provide a necessary and sufficient condition for a plane curve to be perfect. Finally, we show that the equivalent class of generalized asymptotes for a branch of a plane curve can be described as an affine space R m for a certain m. | Asymptotes and perfect curves |
S0167839613001064 | Geometric partial differential equations for curves and surfaces are used in many fields, such as computational geometry, image processing and computer graphics. In this paper, a few differential operators defined on space curves are introduced. Based on these operators, several second-order and fourth-order geometric flows for evolving space curves are constructed. Some properties of the changing rates of the arc-length of the evolved curves and areas swept by the curves are discussed. Short-term and long-term behaviors of the evolved curves are illustrated. | Construction of several second- and fourth-order geometric partial differential equations for space curves |
S0167839614000089 | Splines can be constructed by convolving the indicator function of a cell whose shifts tessellate R n . This paper presents simple, geometric criteria that imply that, for regular shift-invariant tessellations, only a small subset of such spline families yield nested spaces: primarily the well-known tensor-product and box splines. Among the many non-refinable constructions are hex-splines and their generalization to the Voronoi cells of non-Cartesian root lattices. | Refinability of splines derived from regular tessellations |
S0167839614000090 | The problem of designing smoothly rounded right-angle corners with Pythagorean-hodograph (PH) curves is addressed. A G 1 corner can be uniquely specified as a single PH cubic segment, closely approximating a circular arc. Similarly, a G 2 corner can be uniquely constructed with a single PH quintic segment having a unimodal curvature distribution. To obtain G 2 corners incorporating shape freedoms that permit a fine tuning of the curvature profile, PH curves of degree 7 are required. It is shown that degree 7 PH curves define a one-parameter family of G 2 corners, facilitating precise control over the extremum of the unimodal curvature distribution, within a certain range of the parameter. As an alternative, a G 2 corner construction based upon splicing together two PH quintic segments is proposed, that provides two free parameters for shape adjustment. The smooth corner shapes constructed through these schemes can exploit the computational advantages of PH curves, including exact computation of arc length, rational offset curves, and real-time interpolator algorithms for motion control in manufacturing, robotics, inspection, and similar applications. | Construction of G 2 rounded corners with Pythagorean-hodograph curves |
S0167839614000107 | We study a particular class of planar four-bar mechanisms FBM ( Q ) which are based on a given quadrilateral (quad) Q = a 0 a 1 a 2 a 3 . The self-motion of FBM ( Q ) consists of two different parts – one is the motion of an anti-parallelogram whilst the other one is a pure translation with circular paths. We will refer to this translatoric part in this paper only and demonstrate that this translatoric self-motion has the following property: At any moment the positions of the corresponding four coupler points form quads homothetic to Q. This property can be used to define spatial one-parametric motions of an extruded version of the four-bar mechanism which again generate quads of coupler points homothetic to Q. As the next step we take an arbitrary “saturated chain” of quads in space (each vertex shares a vertex with another quad of the set) and define the corresponding one-parametric spatial motions. Then all can be parametrized by the same parameter t. We will show that these partial motions can be interlinked by spherical 2R-joints without locking the one-parametric self-motion. This way the construction delivers a series of new (overconstrained) mechanisms which generalize results on so-called “Fulleroid” linkages. An example based on four quads in space (in planes of a tetrahedron) is worked out in detail. | Overconstrained mechanisms based on planar four-bar-mechanisms |
S0167839614000193 | This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space curves, our method finds the involutions in all cases, and all the rotation symmetries in the particular case of Pythagorean-hodograph curves. Our algorithms solve these problems without converting to implicit form. Instead, we make use of a relationship between two proper parametrizations of the same curve, which leads to algorithms that involve only univariate polynomials. These algorithms have been implemented and tested in the Sage system. | Detecting symmetries of rational plane and space curves |
S0167839614000211 | We provide explicit representations of three moving planes that form a μ-basis for a standard Dupin cyclide. We also show how to compute μ-bases for Dupin cyclides in general position and orientation from their implicit equations. In addition, we describe the role of moving planes and moving spheres in bridging between the implicit and rational parametric representations of these cyclides. | Role of moving planes and moving spheres following Dupin cyclides |
S0167839614000223 | We present an approach to finding the implicit equation of a planar rational parametric cubic curve, by defining a new basis for the representation. The basis, which contains only four cubic bivariate polynomials, is defined in terms of the Bézier control points of the curve. An explicit formula for the coefficients of the implicit curve is given. Moreover, these coefficients lead to simple expressions which describe aspects of the geometric behaviour of the curve. In particular, we present an explicit barycentric formula for the position of the double point, in terms of the Bézier control points of the curve. We also give conditions for when an unwanted singularity occurs in the region of interest. Special cases in which the method fails, such as when three of the control points are collinear, or when two points coincide, will be discussed separately. | A basis for the implicit representation of planar rational cubic Bézier curves |
S0167839614000247 | This paper describes the application of a structure-preserving matrix method to the deconvolution of two Bernstein basis polynomials. Specifically, the deconvolution h ˆ / f ˆ yields a polynomial g ˆ provided the exact polynomial f ˆ is a divisor of the exact polynomial h ˆ and all computations are performed symbolically. In practical situations, however, inexact forms, h and f of, respectively, h ˆ and f ˆ are specified, in which case g = h / f is a rational function and not a polynomial. The simplest method to calculate the coefficients of g is the least squares minimisation of an over-determined system of linear equations in which the coefficient matrix is Tœplitz, but the solution is a polynomial approximation of a rational function. It is shown in this paper that an improved result for g is obtained when the Tœplitz structure of the coefficient matrix is preserved, that is, a structure-preserving matrix method is used. In particular, this method guarantees that a polynomial solution to the deconvolution h / f is obtained, and thus an essential property of the theoretically exact solution is retained in the computed solution. Computational examples that show the improvement in the solution obtained from the structure-preserving matrix method with respect to the least squares solution are presented. | A structure-preserving matrix method for the deconvolution of two Bernstein basis polynomials |
S0167839614000260 | In Winkel (2001) a generalization of Bernstein polynomials and Bézier curves based on umbral calculus has been introduced. In the present paper we describe new geometric and algorithmic properties of this generalization including: (1) families of polynomials introduced by Stancu (1968) and Goldman (1985), i.e., families that include both Bernstein and Lagrange polynomial, are generalized in a new way, (2) a generalized de Casteljau algorithm is discussed, (3) an efficient evaluation of generalized Bézier curves through a linear transformation of the control polygon is described, (4) a simple criterion for endpoint tangentiality is established. | On a generalization of Bernstein polynomials and Bézier curves based on umbral calculus |
S0167839614000272 | We present a method for approximating a point sequence of input points by a -continuous (smooth) arc spline with the minimum number of segments while not exceeding a user-specified tolerance. Arc splines are curves composed of circular arcs and line segments (shortly: segments). For controlling the tolerance we follow a geometric approach: We consider a simple closed polygon P and two disjoint edges designated as the start s and the destination d. Then we compute a SMAP (smooth minimum arc path), i.e. a smooth arc spline running from s to d in P with the minimally possible number of segments. In this paper we focus on the mathematical characterization of possible solutions that enables a constructive approach leading to an efficient algorithm. In contrast to the existing approaches, we do not restrict the breakpoints of the arc spline to a predefined set of points but choose them automatically. This has a considerably positive effect on the resulting number of segments. | Optimal arc spline approximation |
S0167839614000296 | We present a simple method for C-shaped G 2 Hermite interpolation by a rational cubic Bézier curve with conic precision. For the interpolating rational cubic Bézier curve, we derive its control points according to two conic Bézier curves, both matching the G 1 Hermite data and one end curvature of the given G 2 Hermite data, and the weights are obtained by the two given end curvatures. The conic precision property is based on the fact that the two conic Bézier curves are the same when the given G 2 Hermite data are sampled from a conic. Both the control points and weights of the resulting rational cubic Bézier curve are expressed in explicit form. | C-shaped G 2 Hermite interpolation by rational cubic Bézier curve with conic precision |
S0167839614000302 | New bounds on the magnitudes of the first- and second-order partial derivatives of rational triangular Bézier surfaces are presented. Theoretical analysis shows that the proposed bounds are tighter than the existing ones. The superiority of the proposed new bounds is also illustrated by numerical tests. | An improvement on the upper bounds of the magnitudes of derivatives of rational triangular Bézier surfaces |
S0167839614000491 | Salient features in 3D meshes such as small high-curvature details in the middle of largely flat regions are easily ignored by most mesh simplification methods. Nevertheless, these features can be perceived by human observers as perceptually important in CAD models. Recently, mesh saliency has been introduced to identify those visually interesting regions. In this paper, we apply view-based mesh saliency to a purely visual method for surface simplification from two approaches. In the first one, we propose a new simplification error metric that considers polygonal saliency. In the second approach, we use viewpoint saliency as a weighting factor of the quality of a viewpoint in the simplification algorithm. Our results show that saliency can improve the preservation of small but visually significant surfaces even in visual algorithms for surface simplification. However, this comes at a price, because logically some other low-saliency regions in the mesh are simplified further. | Reducing complexity in polygonal meshes with view-based saliency |
S0167839614000521 | We introduce a novel basis for multivariate hierarchical tensor-product spline spaces. Our construction combines the truncation mechanism (Giannelli et al., 2012) with the idea of decoupling basis functions (Mokriš et al., 2014). While the first mechanism ensures the partition of unity property, which is essential for geometric modeling applications, the idea of decoupling allows us to obtain a richer set of basis functions than previous approaches. Consequently, we can guarantee the completeness property of the novel basis for large classes of multi-level spline spaces. In particular, completeness is obtained for the multi-level spline spaces defined on T-meshes for hierarchical splines of (multi-)degree p for example (i) with single knots and p-adic refinement and (ii) with knots of multiplicity m ≥ ( p + 1 ) / 3 and dyadic refinement (where each cell to be refined is subdivided into 2 d cells, with d being the number of variables) without any further restriction on the mesh configuration. Both classes (i), (ii) include multivariate quadratic hierarchical tensor-splines with dyadic refinement. | TDHB-splines: The truncated decoupled basis of hierarchical tensor-product splines |
S0167839614000533 | By means of bond theory, we study Stewart Gough (SG) platforms with n-dimensional self-motions with n > 2 . It turns out that only architecturally singular manipulators can possess these self-motions. Based on this result, we present a complete list of all SG platforms, which have n-dimensional self-motions. Therefore this paper also solves the famous Borel Bricard problem for n-dimensional motions. We also give some remarks and a new result on SG platforms with 2-dimensional self-motions; nevertheless a full discussion of this case remains open. | On Stewart Gough manipulators with multidimensional self-motions |
S0167839614000545 | Ganchev has recently proposed a new approach to minimal surfaces. Introducing canonical principal parameters for these surfaces, he has proved that the normal curvature determines the surface up to its position in the space. Here we prove a theorem that permits to obtain equations of a minimal surface in canonical principal parameters and we make some applications on parametric polynomial minimal surfaces. Thus we show that Ganchev's approach implies an effective method to prove the coincidence of two minimal surfaces given in isothermal coordinates by different parametric equations. | Transition to canonical principal parameters on minimal surfaces |
S0167839614000570 | A rational curve on a rational surface such that the unit normal vector field of the surface along this curve is rational will be called a curve providing Pythagorean surface normals (or shortly a PSN curve). These curves represent rational paths on the surface along which the surface possesses rational offset curves. Our aim is to study rational surfaces containing enough PSN curves. The relation with PN surfaces will be also investigated and thoroughly discussed. The algebraic and geometric properties of PSN curves will be described using the theory of double planes. The main motivation for this contribution is to bring the theory of rational offsets of rational surfaces closer to the practical problems appearing in numerical-control machining where the milling cutter does not follow continuously the whole offset surface but only certain chosen trajectories on it. A special attention will be devoted to rational surfaces with pencils of PSN curves. | Surfaces with Pythagorean normals along rational curves |
S0167839614000582 | In this work, we propose a structured computational framework for modelling the envelope of the swept volume, that is the boundary of the volume obtained by sweeping an input solid along a trajectory of rigid motions. Our framework is adapted to the well-established industry-standard brep format to enable its implementation in modern CAD systems. This is achieved via a “local analysis”, which covers parametrizations and singularities, as well as a “global theory” which tackles face-boundaries, self-intersections and trim curves. Central to the local analysis is the “funnel” which serves as a natural parameter space for the basic surfaces constituting the sweep. The trimming problem is reduced to the problem of surface–surface intersections of these basic surfaces. Based on the complexity of these intersections, we introduce a novel classification of sweeps as decomposable and non-decomposable. Further, we construct an invariant function θ on the funnel which efficiently separates decomposable and non-decomposable sweeps. Through a geometric theorem we also show intimate connections between θ, local curvatures and the inverse trajectory used in earlier works as an approach towards trimming. In contrast to the inverse trajectory approach of testing points, θ is a computationally robust global function. It is the key to a complete structural understanding, and an efficient computation of both, the singular locus and the trim curves, which are central to a stable implementation. Several illustrative outputs of a pilot implementation are included. | Local and global analysis of parametric solid sweeps |
S0167839614000600 | In this paper, we introduce triangular subdivision operators which are composed of a refinement operator and several averaging operators, where the refinement operator splits each triangle uniformly into four congruent triangles and in each averaging operation, every vertex will be replaced by a convex combination of itself and its neighboring vertices. These operators form an infinite class of triangular subdivision schemes including Loop's algorithm with a restricted parameter range and the midpoint schemes for triangular meshes. We analyze the smoothness of the resulting subdivision surfaces at their regular and extraordinary points by generalizing an established technique for analyzing midpoint subdivision on quadrilateral meshes. General triangular midpoint subdivision surfaces are smooth at all regular points and they are also smooth at extraordinary points under certain conditions. We show some general triangular subdivision surfaces and compare them with Loop subdivision surfaces. | General triangular midpoint subdivision |
S0167839614000636 | We consider isogeometric functions and their derivatives. Given a geometry mapping, which is defined by an n-dimensional NURBS patch in R d , an isogeometric function is obtained by composing the inverse of the geometry mapping with a NURBS function in the parameter domain. Hence an isogeometric function can be represented by a NURBS parametrization of its graph. We take advantage of the projective representation of the NURBS patch as a B-spline patch in homogeneous coordinates. We derive a closed form representation of the graph of a partial derivative of an isogeometric function. The derivative can be interpreted as an isogeometric function of higher degree and lower smoothness on the same piecewise rational geometry mapping, hence the space of isogeometric functions is closed under differentiation. We distinguish the two cases n = d and n < d , with a focus on n = d − 1 in the latter one. As a first application of the presented formula we derive conditions which guarantee C 1 and C 2 smoothness for isogeometric functions on several singularly parametrized planar and volumetric domains as well as on embedded surfaces. It is interesting to note that the presented conditions depend not only on the general structure of the patch, but on the exact representation of the interior of the given geometry mapping. | Derivatives of isogeometric functions on n-dimensional rational patches in R d |
S0167839614000648 | One major issue in CAGD is to model complex objects using free-form surfaces of general topology. A natural approach is curvenet-based design, where designers directly create and modify feature curves. These are interpolated by smoothly connected, multi-sided patches, which can be represented by transfinite surfaces, defined as a combination of side interpolants or ribbons. A ribbon embeds Hermite data, i.e., prescribed positional and cross-derivative functions along boundary curves. The paper focuses on two transfinite schemes: the first is an enhanced and extended variant of a multi-sided generalization of the classical Coons patch (Várady et al., 2011); the second one is based on a new concept of combining doubly curved composite ribbons, each one interpolating three adjacent sides. Main contributions include various ribbon parameterizations that surpass former methods in quality and computational efficiency. It is proven that these surfaces smoothly interpolate the prescribed ribbon data. Both formulations are based on non-regular convex polygonal domains and distance-based, rational blending functions. A few examples illustrate the results. | Ribbon-based transfinite surfaces |
S0167839614000752 | For an arbitrary degree Bézier curve B , we first establish sufficient conditions for its control polygon to become homeomorphic to B via subdivision. This is extended to show a subdivided control polygon that is ambient isotopic to B . We provide closed-form formulas to compute the corresponding number of iterations for equivalence under homeomorphism and ambient isotopy. The development of these a priori values was motivated by application to high performance computing (HPC), where providing estimates of total run time is important for scheduling. | Isotopic equivalence by Bézier curve subdivision for application to high performance computing |
S0167839614000764 | This paper presents a biarc-based subdivision scheme for space curve interpolation. Given a sequence of space points, or a sequence of space points and tangent vectors, the scheme produces a smooth curve interpolating all input points by iteratively inserting new points and computing new tangent vectors. For each step of subdivision, the newly inserted point corresponding to an existing edge is a specified joint point of a biarc curve which interpolates the two end points and the tangents. A provisional tangent is also assigned to the newly inserted point. Each of the tangents for the following subdivision step is further updated as a linear blending of the provisional tangent and the tangent at the respective point of a circle passing through the local point and its two adjacent points. If adjacent four initial points and their initial reference tangent vectors are sampled from a spherical surface, the limit curve segment between the middle two initial points exactly lies on the same spherical surface. The biarc based subdivision scheme is proved to be G 1 continuous with a nice convexity preserving property. Numerical examples also show that the limit curves are G 2 continuous and fair. Several examples are given to demonstrate the excellent properties of the scheme. | A biarc based subdivision scheme for space curve interpolation |
S0167839614000806 | In this paper two kinds of bivariate S-λ basis functions, tensor product S-λ basis functions and triangular S-λ basis functions, are constructed by means of the technique of generating functions and transformation factors. These two kinds of bivariate S-λ basis functions have lots of important properties, such as non-negativity, partition of unity, linear independence and so on. The framework of the tensor product S-λ basis functions provides a unified scheme for dealing with several kinds of tensor product basis functions, such as the tensor product Bernstein basis functions, the tensor product Poisson basis functions and some other new tensor product basis functions. The framework of the triangular S-λ surface basis functions includes the triangular Bernstein basis functions, the rational triangular Bernstein basis functions and some other new triangular basis functions. Moreover, the corresponding two kinds of S-λ surfaces are constructed by means of these two kinds of bivariate basis functions, respectively. These two kinds of S-λ surface patches have the important properties of surface modeling, such as affine invariance, convex hull property and so on. | Bivariate S-λ bases and S-λ surface patches |
S0167839614000818 | We show how to find four generic interpolants to a C 1 Hermite data-set in the complex representation, using Pythagorean-hodograph curves generated as cuts of degree ( 1 , 3 ) of Laurent series. The developed numerical experiments have shown that two of these interpolants are simple curves and that these (at least) have stable shape, in the sense that their topologies persist when the direction of the velocity at each end-point changes. Our curves are fair, but have different shapes to those of other interpolants. Unlike existing methods, our technique allows regular PH interpolants to be found for special collinear C 1 Hermite data-sets. | Planar C 1 Hermite interpolation with PH cuts of degree ( 1 , 3 ) of Laurent series |
S0167839614000983 | The purpose of this paper is to present algorithms for computing all the differential geometry properties of non-transversal intersection curves of three parametric hypersurfaces in Euclidean 4-space. For transversal intersections, the tangential direction at an intersection point can be computed by the extension of the vector product of the normal vectors of three hypersurfaces. However, when the three normal vectors are not linearly independent, the tangent direction cannot be determined by this method. If normal vectors of hypersurfaces are parallel ( N 1 = N 2 = N 3 ) we have tangential intersection, and if normal vectors of hypersurfaces are not parallel but are linearly dependent we have “almost tangential” intersection. In each case, we obtain unit tangent vector (t), principal normal vector (n), binormal vectors ( b 1 , b 2 ) and curvatures ( k 1 , k 2 , k 3 ) of the intersection curve. | Differential geometry of non-transversal intersection curves of three parametric hypersurfaces in Euclidean 4-space |
S0167839614000995 | A swept surface is generated from a profile curve and a sweep curve by employing the latter to define a continuous family of transformations of the former. By using polynomial or rational curves, and specifying the homogeneous coordinates of the swept surface as bilinear forms in the profile and sweep curve homogeneous coordinates, the outcome is guaranteed to be a rational surface compatible with the prevailing data types of CAD systems. However, this approach does not accommodate many geometrically intuitive sweep operations based on differential or integral properties of the sweep curve — such as the parametric speed, tangent, normal, curvature, arc length, and offset curves — since they do not ordinarily have a rational dependence on the curve parameter. The use of Pythagorean-hodograph (PH) sweep curves surmounts this limitation, and thus makes possible a much richer spectrum of rational swept surface types. A number of representative examples are used to illustrate the diversity of these novel swept surface forms — including the oriented-translation sweep, offset-translation sweep, generalized conical sweep, and oriented-involute sweep. In many cases of practical interest, these forms also have rational offset surfaces. Considerations related to the automated CNC machining of these surfaces, using only their high-level procedural definitions, are also briefly discussed. | Rational swept surface constructions based on differential and integral sweep curve properties |
S0167839614001009 | Fitting a sparse surface to approximate vast dense data is of interest for many applications: reverse engineering, recognition and compression, etc. The present work provides an approach to fit a Loop subdivision surface to a dense triangular mesh of arbitrary topology, whilst preserving and aligning the original features. The natural ridge-joined connectivity of umbilics and ridge-crossings is used as the connectivity of the control mesh for subdivision, so that the edges follow salient features on the surface. Furthermore, the chosen features and connectivity characterise the overall shape of the original mesh, since ridges capture extreme principal curvatures and ridges start and end at umbilics. A metric of Hausdorff distance including curvature vectors is proposed and implemented in a distance transform algorithm to construct the connectivity. Ridge-colour matching is introduced as a criterion for edge flipping to improve feature alignment. Several examples are provided to demonstrate the feature-preserving capability of the proposed approach. | Subdivision surface fitting to a dense mesh using ridges and umbilics |
S0167839614001010 | A generic planar quadrilateral defines a 2:1 bilinear map. We show that by assigning an appropriate weight to one vertex of any planar quadrilateral, we can create a map whose inverse is rational linear. | Birational quadrilateral maps |
S0167839614001022 | In this work we present a parameter-dependent Refine-and-Smooth (RS) subdivision algorithm where the refine stage R consists in the application of a perturbation of Chaikin's/Doo–Sabin's vertex split, while each smoothing stage S performs averages of adjacent vertices like in the Lane–Riesenfeld algorithm (Lane and Riesenfeld, 1980). This constructive approach provides a unifying framework for univariate/bivariate primal and dual subdivision schemes with tension parameter and allows us to show that several existing subdivision algorithms, proposed in the literature via isolated constructions, can be obtained as specific instances of the proposed strategy. Moreover, this novel approach provides an intuitive theoretical tool for the derivation of new non-tensor product subdivision schemes that in the regular regions satisfy the property of reproducing bivariate cubic polynomials, thus resulting the natural extension of the univariate family presented in Hormann and Sabin (2008). | A Chaikin-based variant of Lane–Riesenfeld algorithm and its non-tensor product extension |
S0167839614001113 | Recently, it was shown that a bi-cubic patch complex with n-sided holes can be completed into a curvature-continuous () surface by n-sided caps of degree bi-5 that offer good and flexible shape (Karciauskas and Peters, 2013). This paper further explores the space of n-sided caps of degree bi-5 but focuses on functionals to set degrees of freedom and to optimally propagate and average out curvature from the bi-cubic complex. | Biquintic G 2 surfaces via functionals |
S0167839614001125 | The μ -invariant μ = ( μ 1 , μ 2 , μ 3 ) of a rational space curve gives important information about the curve. In this paper, we describe the structure of all parameterizations that have the same μ -type, what we call a μ-stratum, and as well the closure of strata. Many of our results are based on papers by the second author that appeared in the commutative algebra literature. We also present new results, including an explicit formula for the codimension of the locus of non-proper parametrizations within each μ -stratum and a decomposition of the smallest μ -stratum based on which two-dimensional rational normal scroll the curve lies on. | Strata of rational space curves |
S0167839615000023 | In this study, we propose a robust algorithm for reconstructing free-form space curves in space using a Non-Uniform Rational B-Spline (NURBS)-snake model. Two perspective images of the required free-form curve are used as the input and a nonlinear optimization process is used to fit a NURBS-snake on the projected data in these images. Control points and weights are treated as decision variables in the optimization process. The Levenberg–Marquardt optimization algorithm is used to optimize the parameters of the NURBS-snake, where the initial solution is obtained using a two-step procedure. This makes the convergence faster and it stabilizes the optimization procedure. The curve reconstruction problem is reduced to a problem that comprises stereo reconstruction of the control points and computation of the corresponding weights. Several experiments were conducted to evaluate the performance of the proposed algorithm and comparisons were made with other existing approaches. | Reconstruction of free-form space curves using NURBS-snakes and a quadratic programming approach |
S0167839615000138 | We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly faster, simpler, and more general than an earlier method addressing a similar problem (Alcázar et al., 2014b). To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage. | Symmetry detection of rational space curves from their curvature and torsion |
S0167839615000151 | Ck (geometrically continuous surface) constructions were developed to create surfaces that are smooth also at irregular points where, in a quad-mesh, three or more than four elements come together. Isogeometric elements were developed to unify the representation of geometry and of engineering analysis. We show how matched constructions for geometry and analysis automatically yield isogeometric elements. This provides a formal framework for the existing and any future isogeometric elements based on geometric continuity. | Matched G k -constructions always yield C k -continuous isogeometric elements |
S0167839615000163 | We present an efficient adaptive refinement procedure that preserves analysis-suitability of the T-mesh, that is, the linear independence of the T-spline blending functions. We prove analysis-suitability of the overlays and boundedness of their cardinalities, nestedness of the generated T-spline spaces, and linear computational complexity of the refinement procedure in terms of the number of marked and generated mesh elements. | Analysis-suitable adaptive T-mesh refinement with linear complexity |
S0167839615000400 | Toric surface patches are a multi-sided generalization of classical rational Bézier surface patches which are widely used in free-form surface modeling. In this paper, we present the first derivatives of toric surface patches along the boundary and study the G 1 continuity between adjacent toric surface patches by the toric degenerations. Furthermore, some practical G 1 sufficient conditions of toric surface patches are developed and the representative examples are given. | G 1 continuity between toric surface patches |
S0167839615000448 | The purpose of this article is the construction of a normalized basis for a quadratic condensed Powell–Sabin-12 macro-element space introduced by Alfeld et al. (2010). The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction of this basis is adopted from Dierckx (1997) and Speleers (2010a), and is based on the determination of a set of triangles that must contain a specific set of points. The proposed basis can only be constructed on triangulations with a maximal angle less than π 2 . | A normalized basis for condensed C 1 Powell–Sabin-12 splines |
S0167839615000540 | The investigation of the umbral calculus based generalization of Bernstein polynomials and Bézier curves is continued in this paper: First a generalization of the de Casteljau algorithm that uses umbral shift operators is described. Then it is shown that the quite involved umbral shifts can be replaced by a surprisingly simple recursion which in turn can be understood in geometrical terms as an extension of the de Casteljau interpolation scheme. Namely, instead of using only the control points of level r − 1 to generate the points on level r as in the ordinary de Casteljau algorithm, one uses also points on level r − 2 or more previous levels. Thus the unintuitive parameters in the algebraic definition of generalized Bernstein polynomials get geometric meaning. On this basis a new direct method for the design of Bézier curves is described that allows to adapt the control polygon as a whole by moving a point of the associated Bézier curve. | On a generalization of Bernstein polynomials and Bézier curves based on umbral calculus (II): de Casteljau algorithm |
S0167839615000783 | We consider a C 1 cubic spline space defined over a triangulation with Powell–Sabin refinement. The space has some local C 2 super-smoothness and can be seen as a close extension of the classical cubic Clough–Tocher spline space. In addition, we construct a suitable normalized B-spline representation for this spline space. The basis functions have a local support, they are nonnegative, and they form a partition of unity. We also show how to compute the Bézier control net of such a spline in a stable way. | A new B-spline representation for cubic splines over Powell–Sabin triangulations |
S0167839615000795 | The boundary representations (B-reps) that are used to represent shape in Computer-Aided Design systems create unavoidable gaps at the face boundaries of a model. Although these inconsistencies can be kept below the scale that is important for visualisation and manufacture, they cause problems for many downstream tasks, making it difficult to use CAD models directly for simulation or advanced geometric analysis, for example. Motivated by this need for watertight models, we address the problem of converting B-rep models to a collection of cubic C 1 Clough–Tocher splines. These splines allow a watertight join between B-rep faces, provide a homogeneous representation of shape, and also support local adaptivity. We perform a comparative study of the most prominent Clough–Tocher constructions and include some novel variants. Our criteria include visual fairness, invariance to affine reparameterisations, polynomial precision and approximation error. The constructions are tested on both synthetic data and CAD models that have been triangulated. Our results show that no construction is optimal in every scenario, with surface quality depending heavily on the triangulation and parameterisation that are used. | Watertight conversion of trimmed CAD surfaces to Clough–Tocher splines |
S0167839615000801 | We describe a construction of LR-spaces whose bases are composed of locally linearly independent B-splines which also form a partition of unity. The construction conforms to given refinement requirements associated to subdomains. In contrast to the original LR-paper (Dokken et al., 2013) and similarly to the hierarchical B-spline framework (Forsey and Bartels, 1988) the construction of the mesh is based on a priori choice of a sequence of nested tensor B-spline spaces. | A hierarchical construction of LR meshes in 2D |
S0167839615000813 | We introduce and analyze univariate, linear, and stationary subdivision schemes for refining noisy data by fitting local least squares polynomials. This is the first attempt to design subdivision schemes for noisy data. We present primal schemes, with refinement rules based on locally fitting linear polynomials to the data, and study their convergence, smoothness, and basic limit functions. Then, we provide several numerical experiments that demonstrate the limit functions generated by these schemes from initial noisy data. The application of an advanced local linear regression method to the same data shows that the methods are comparable. In addition, several extensions and variants are discussed and their performance is illustrated by examples. We conclude by applying the schemes to noisy geometric data. | Univariate subdivision schemes for noisy data with geometric applications |
S0167839615000825 | A new equivalence notion between non-stationary subdivision schemes, termed asymptotic similarity, which is weaker than asymptotic equivalence, is introduced and studied. It is known that asymptotic equivalence between a non-stationary subdivision scheme and a convergent stationary scheme guarantees the convergence of the non-stationary scheme. We show that for non-stationary schemes reproducing constants, the condition of asymptotic equivalence can be relaxed to asymptotic similarity. This result applies to a wide class of non-stationary schemes. | Convergence of univariate non-stationary subdivision schemes via asymptotic similarity |
S0167839615000837 | We introduce the concept of reduced curvature formulae for 3-D space entities (surfaces, curves). A reduced formula entails only derivatives of the functions involved in the entity's representation and admits no further algebraic simplifications. Although not always the most compact, reduced curvature formulae entail only basic arithmetic operators and are more efficient computationally compared to alternative unreduced formulae. Reduced formulae are presented for the normal, mean and Gaussian curvatures of a surface and the curvature of curves on a surface, where each surface or curve on a surface may be defined parametrically or implicitly. Reduced formulae are also presented for the curvature of surface intersection curves, where each of the intersecting surfaces may be a given surface or an offset of a given surface and each given surface may be defined parametrically or implicitly. Known formulae are cited, without derivation, to form a collection, in one place, of new and of known results scattered in the literature. Each curve curvature formula is presented together with a formula for the respective binormal vector, from which formulae for the Frenet frame and torsion of the curve can be derived. | Reduced curvature formulae for surfaces, offset surfaces, curves on a surface and surface intersections |
S0167839615000849 | We provide a simple, efficient technique for computing μ-bases for quadric surfaces from their rational quadratic parametrizations. Our major innovation is to simplify the computations by using complex parameters, even though all the surfaces we treat have only real coefficients in both their implicit and parametric representations. In addition to the theory, we provide several examples to illustrate our method. | Complex μ-bases for real quadric surfaces |
S0167839615000850 | In this article we provide the characterization of analysis suitable T-spline spaces (Beirão da Veiga et al., 2013) as the space of piecewise polynomials with appropriate linear constrains on the subdomain interfaces. We describe AST-meshes for which the linear constrains are equivalent to smoothness conditions and provide examples showing that this is not always the case. | Characterization of analysis-suitable T-splines |
S0167839615000941 | Optimal recursive decomposition (or DR-planning) is crucial for analyzing, designing, solving or finding realizations of geometric constraint systems. While the optimal DR-planning problem is NP-hard even for (general) 2D bar–joint constraint systems, we describe an O ( n 3 ) algorithm for a broad class of constraint systems that are isostatic or underconstrained. The algorithm achieves optimality by using the new notion of a canonical DR-plan that also meets various desirable, previously studied criteria. In addition, we leverage recent results on Cayley configuration spaces to show that the indecomposable systems – that are solved at the nodes of the optimal DR-plan by recombining solutions to child systems – can be minimally modified to become decomposable and have a small DR-plan, leading to efficient realization algorithms. We show formal connections to well-known problems such as completion of underconstrained systems. Well suited to these methods are classes of constraint systems that can be used to efficiently model, design and analyze quasi-uniform (aperiodic) and self-similar, layered material structures. We formally illustrate by modeling silica bilayers as body–hyperpin systems and cross-linking microfibrils as pinned line-incidence systems. A software implementation of our algorithms and videos demonstrating the software are publicly available online (visit http://cise.ufl.edu/~tbaker/drp/index.html). | Optimal decomposition and recombination of isostatic geometric constraint systems for designing layered materials |
S0167839615000953 | In this paper, we discuss the bicubic C 2 spline spaces over hierarchical T-meshes in detail. The topological explanation of the dimension formula is further explored. We provide three necessary conditions for the completeness of the spline spaces. Based on the three conditions, the rule of the refinement of the hierarchical T-mesh is given, and a basis is constructed. The basis functions are linearly independent and complete, which provides a solution for the defects in the former literature. | Bicubic hierarchical B-splines: Dimensions, completeness, and bases |
S0167839615000965 | This paper proposes a geometric algorithm for computation of geodesic on surfaces. The geodesics on surfaces are traced in a simple way which is independent of the complex description of the geodesic equations. Through derivation process, the calculation error of this algorithm is obtained. A step size adjustment strategy which enables the step size adapt to the geometry of surface is introduced. The proposed method is also compared to some other well-known methods in this study. Many geodesics computed using these approaches on various B-spline surfaces or their equivalent tessellated surfaces have been presented. Experiments demonstrate that the proposed algorithm is efficient. Meanwhile, the results show that the step size adjustment strategy works well for most of the cases. | A geometric method for computation of geodesic on parametric surfaces |
S0167839615000977 | Many problems in computer aided geometric design and computer graphics can be turned into a root-finding problem of a polynomial equation. Among various solutions, clipping methods based on the Bernstein–Bézier form usually have good numerical stability. A traditional clipping method using polynomials of degree r can achieve a convergence rate of r + 1 for a single root. It utilizes two polynomials of degree r to bound the given polynomial f ( t ) of degree n, where r = 2 , 3 , and the roots of the bounding polynomials are used for clipping off the subintervals containing no roots of f ( t ) . This paper presents a rational cubic clipping method for finding the roots of a polynomial f ( t ) within an interval. The bounding rational cubics can achieve an approximation order of 7 and the corresponding convergence rate for finding a single root is also 7. In addition, differently from the traditional cubic clipping method solving the two bounding polynomials in O ( n 2 ) , the new method directly constructs the two rational cubics in O ( n ) which can be used for bounding f ( t ) in many cases. Some examples are provided to show the efficiency, the approximation effect and the convergence rate of the new method. | A rational cubic clipping method for computing real roots of a polynomial |
S0167839615001004 | The Laplace–Beltrami operator is the foundation of describing geometric partial differential equations, and it also plays an important role in the fields of computational geometry, computer graphics and image processing, such as surface parameterization, shape analysis, matching and interpolation. However, constructing the discretized Laplace–Beltrami operator with convergent property has been an open problem. In this paper we propose a new discretization scheme of the Laplace–Beltrami operator over triangulated surfaces. We prove that our discretization of the Laplace–Beltrami operator converges to the Laplace–Beltrami operator at every point of an arbitrary smooth surface as the size of the triangular mesh over the surface tends to zero. Numerical experiments are conducted, which support the theoretical analysis. | Localized discrete Laplace–Beltrami operator over triangular mesh |
S0167839615001016 | This paper presents a novel 3D shape descriptor which explicitly captures the local geometry and encodes it using an efficient representation. Our method consists of multiple evolving fronts which are realized by a set of growing spheres on the surface. At the core of this method is a simple intersection operator between the spheres and the shape's surface. Intersection curves yield a discrete sampling of the surface at different positions and scales. Our key idea is to define a shape descriptor that captures the continuous local geometry of the surface in an efficient and consistent representation by intersecting the surface with multiple spheres and transforming the intersection curve to frequency domain. To evaluate our descriptor, we define shape similarity metric and perform shape matching on the SHREC11 non-rigid benchmark and other classes. | Sphere intersection 3D shape descriptor (SID) |
S0167839615001028 | In this paper we investigate the problem of interpolating a B-spline curve network, in order to create a surface satisfying such a constraint and defined by blending functions spanning the space of bivariate C 1 quadratic splines on criss–cross triangulations. We prove the existence and uniqueness of the surface, providing a constructive algorithm for its generation. We also present numerical and graphical results and comparisons with other methods. | Curve network interpolation by C 1 quadratic B-spline surfaces |
S0167839615001041 | Extended Chebyshev spaces that also comprise the constants represent large families of functions that can be used in real-life modeling or engineering applications that also involve important (e.g. transcendental) integral or rational curves and surfaces. Concerning CAGD, the unique normalized B-bases of such vector spaces ensure optimal shape preserving properties, important evaluation or subdivision algorithms and useful shape parameters. Therefore, we propose global explicit formulas for the entries of those transformation matrices that map these normalized B-bases to the traditional (or ordinary) bases of the underlying vector spaces. Then, we also describe general and ready to use control point configurations for the exact representation of those traditional integral parametric curves and (hybrid) surfaces that are specified by coordinate functions given as (products of separable) linear combinations of ordinary basis functions. The obtained results are also extended to the control point and weight based exact description of the rational counterpart of these integral parametric curves and surfaces. The universal applicability of our methods is presented through polynomial, trigonometric, hyperbolic or mixed extended Chebyshev vector spaces. | Control point based exact description of curves and surfaces, in extended Chebyshev spaces |
S0167839615001193 | We propose a general scheme for detecting critical locations (of dimension zero) of piecewise polynomial multivariate equation systems. Our approach generalizes previously known methods for locating tangency events or self-intersections, in contexts such as surfaceâÂÂsurface intersection (SSI) problems and the problem of tracing implicit plane curves. Given the algebraic constraints of the original problem, we formulate additional constraints, seeking locations where the differential matrix of the original problem has a non-maximal rank. This makes the method independent of a specific geometric application, as well as of dimensionality. Within the framework of subdivision based solvers, test results are demonstrated for non-linear systems with three and four unknowns. | Detection of critical points of multivariate piecewise polynomial systems |
S0167839615001211 | In this paper we devise a new algorithm for completing surface with missing geometry and topology founded upon the theory and techniques of sparse signal recovery. The key intuition is that any meaningful 3D shape, represented as a discrete mesh, almost certainly possesses a low-dimensional intrinsic structure, which can be expressed as a sparse representation in some transformed domains. Instead of estimating the missing geometry directly, our novel method is to find this low-dimensional representation which describes the entire original shape. More specifically, we find that, for many shapes, the vertex coordinate function can be well approximated by a very sparse coefficient representation with respect to the dictionary comprising its Laplacian eigenfunctions, and it is then possible to recover this sparse representation from partial measurements of the original shape. Taking advantage of the sparsity cue, we advocate a novel variational approach for surface inpainting, integrating data fidelity constraints on the shape domain with coefficient sparsity constraints on the transformed domain. Because of the powerful properties of Laplacian eigenbasis, the inpainting results of our method tend to be smooth and globally coherent with the remaining shape. We demonstrate the performance of our new method via various examples in geometry restoration, shape repair, and hole filling. | Surface inpainting with sparsity constraints |
S0167839615001223 | In the paper two new approaches for construction of parametric polynomial approximants of a unit circle are presented. The obtained approximants have better approximation properties than those given by other methods, i.e., smaller radial error, symmetry, and exponential error decay. | Uniform approximation of a circle by a parametric polynomial curve |
S0167839615001351 | We derive explicit formulas for the μ-bases of conic sections and planar rational cubic curves. Using the μ-bases for planar rational cubic curves, we find explicit formulas for their implicit equations and double points. We also extend the explicit formula for the μ-bases of conic sections to μ-bases for rational curves of degree n in n-dimensions. | Explicit μ-bases for conic sections and planar rational cubic curves |
S0167839615001363 | We consider the adaptive refinement of bivariate quartic C 2 -smooth box spline spaces on the three-directional (type-I) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quartic polynomials, which will be called the space of special quartics. Given a bounded domain Ω ⊂ R 2 and finite sequence ( G ℓ ) ℓ = 0 , … , N of dyadically refined grids, we obtain a hierarchical grid by selecting mutually disjoint cells from all levels such that their union covers the entire domain. Using a suitable selection procedure allows to define a basis spanning the hierarchical box spline space. The paper derives a characterization of this space. Under certain mild assumptions on the hierarchical grid, the hierarchical spline space is shown to contain all C 2 -smooth functions whose restrictions to the cells of the hierarchical grid are special quartic polynomials. Thus, in this case we can give an affirmative answer to the completeness questions for the hierarchical box spline basis. | Characterization of bivariate hierarchical quartic box splines on a three-directional grid |
S0167839615001375 | Rational curves and surfaces are powerful tools for shape representation and geometric modeling. However, the real weights are generally difficult to choose except for a few special cases such as representing conics. This paper presents an extension of rational curves and surfaces by replacing the real weights with matrices. The matrix weighted rational curves and surfaces have the same structures as the traditional rational curves and surfaces but the matrix weights can be defined in geometric ways. In particular, the weight matrices for the extended rational Bézier, NURBS or the generalized subdivision curves and surfaces are computed using the normal vectors specified at the control points. Similar to the effects of control points, the specified normals can be used to control the curve or the surface's shape efficiently. It is also shown that matrix weighted NURBS curves and surfaces can pass through their control points, thus curve or surface reconstruction by the extended NURBS model no longer needs solving any large system but just choosing control points and control normals from the input data. | Matrix weighted rational curves and surfaces |
S0167839615001387 | This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject. The goal of this survey paper is to describe how splines arise in geometry and topology. Geometric splines usually appear under the name GKM theory after GoreskyâÂÂKottwitzâÂÂMacPherson, who developed them to compute what is called the cohomology ring of a geometric object. Geometers and analysts ask many of the same questions about splines: what is their dimension? can we identify a basis? can we find explicit formulas for the elements of the basis? However geometric constraints can change the tone of these questions: the splines may satisfy various symmetries or have a basis satisfying certain conditions. And some questions are specific to geometric splines: geometers particularly care about the multiplication table with respect to a given basis. | Splines in geometry and topology |
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