Constrained multi-degree reduction of rational Bézier curves using reparameterization

Applying homogeneous coordinates, we extend a newly appeared algorithm of best constrained multi-degree reduction for polynomial Bezier curves to the algorithms of constrained multi-degree reduction for rational Bezier curves. The idea is introducing two criteria, variance criterion and ratio criterion, for reparameterization of rational Bezier curves, which are used to make uniform the weights of the rational Bezier curves as accordant as possible, and then do multi-degree reduction for each component in homogeneous coordinates. Compared with the two traditional algorithms of ＂cancelling the best linear common divisor＂ and ＂shifted Chebyshev polynomial＂, the two new algorithms presented here using reparameterization have advantages of simplicity and fast computing, being able to preserve high degrees continuity at the end points of the curves, do multi-degree reduction at one time, and have good approximating effect.

A NEW METHOD FOR SOLUTING TRANSIENT RESPONSE OF GENERALLY VISCOUS DAMPING MULTI-DEGREE SYSTEM

By means of the theory of composite-modality,the superposition principle of the vibra-tion mode of the linear system,and the analytical method of the original coordinate,a mathematical model of transient response to any stimulus for generally viscous damping multi-degree system was established.This method not only solves the problem of the transient response of displacement,but also calculates the transient response of the elastic force or the elastic couple of the system.

Multi-degree reduction of tensor product Bézier surfaces with conditions of corners interpolations

This paper studies the multi-degree reduction of tensor product B(?)zier surfaces with any degree interpolation conditions of four corners, which is urgently to be resolved in many CAD/CAM systems. For the given conditions of corners interpolation, this paper presents one intuitive method of degree reduction of parametric surfaces. Another new approximation algorithm of multi-degree reduction is also presented with the degree elevation of surfaces and the Chebyshev polynomial approximation theory. It obtains the good approximate effect and the boundaries of degree reduced surface can preserve the prescribed continuities. The degree reduction error of the latter algorithm is much smaller than that of the first algorithm. The error bounds of degree reduction of two algorithms are also presented .

Multi-degree reduction of NURBS curves based on their explicit matrix representation and polynomial approximation theory

NURBS curve is one of the most commonly used tools in CAD systems and geometric modeling for its various specialties, which means that its shape is locally adjustable as well as its continuity order, and it can represent a conic curve precisely. But how to do degree reduction of NURBS curves in a fast and efficient way still remains a puzzling problem. By applying the theory of the best uniform approximation of Chebyshev polynomials and the explicit matrix representation of NURBS curves, this paper gives the necessary and sufficient condition for degree reducible NURBS curves in an explicit form. And a new way of doing degree reduction of NURBS curves is also presented, including the multi-degree reduction of a NURBS curve on each knot span and the multi-degree reduction of a whole NURBS curve. This method is easy to carry out, and only involves simple calculations. It provides a new way of doing degree reduction of NURBS curves, which can be widely used in computer graphics and industrial design.

Optimal constrained multi-degree reduction of Bézier curves with explicit expressions based on divide and conquer

We decompose the problem of the optimal multi-degree reduction of Bézier curves with corners constraint into two simpler subproblems, namely making high order interpolations at the two endpoints without degree reduction, and doing optimal degree reduction without making high order interpolations at the two endpoints. Further, we convert the second subproblem into multi-degree reduction of Jacobi polynomials. Then, we can easily derive the optimal solution using orthonormality of Jacobi polynomials and the least square method of unequally accurate measurement. This method of 'divide and conquer' has several advantages including maintaining high continuity at the two endpoints of the curve, doing multi-degree reduction only once, using explicit approximation expressions, estimating error in advance, low time cost, and high precision. More importantly, it is not only deduced simply and directly, but also can be easily extended to the degree reduction of surfaces. Finally, we present two examples to demonstrate the effectiveness of our algorithm.

A novel algorithm for explicit optimal multi-degree reduction of triangular surfaces

This paper introduces the algebraic property of bivariate orthonormal Jacobi polynomials into geometric approximation. Based on the latest results on the transformation formulae between bivariate Bernstein polynomials and Jacobi polynomials, we naturally deduce a novel algorithm for multi-degree reduction of triangular B^zier surfaces. This algorithm possesses four characteristics： ability of error forecast, explicit expression, less time consumption, and best precision. That is, firstly, whether there exists a multi-degree reduced surface within a prescribed tolerance is judged beforehand; secondly, all the operations of multi-degree reduction are just to multiply the column vector generated by sorting the series of the control points of the original surface in lexicographic order by a matrix; thirdly, this matrix can be computed at one time and stored in an array before processing degree reduction; fourthly, the multi-degree reduced surface achieves an optimal approximation in the norm L2. Some numerical experiments are presented to validate the effectiveness of this algorithm, and to show that the algorithm is applicable to information processing of products in CAD system.

A Geometric Approach for Multi-Degree Spline

Multi-degree spline （MD-spline for short） is a generalization of B-spline which comprises of polynomial segments of various degrees. The present paper provides a new definition for MD-spline curves in a geometric intuitive way based on an efficient and simple evaluation algorithm. MD-spline curves maintain various desirable properties of B-spline curves, such as convex hull, local support and variation diminishing properties. They can also be refined exactly with knot insertion. The continuity between two adjacent segments with different degrees is at least C1 and that between two adjacent segments of same degrees d is C^d-1. Benefited by the exact refinement algorithm, we also provide several operators for MD-spline curves, such as converting each curve segment into Bezier form, an efficient merging algorithm and a new curve subdivision scheme which allows different degrees for each segment.

ArtiSentialTM Revolutionizes Traditional Laparoscopy in Urology by Introducing the Dexterity of a Robotic System to a Laparoscopic Instrument —First Experiences in Urology in Europe by the Ukrb in Neuruppin

We describe the first two conventional, laparoscopic renal operations with a new multi-degree of freedom articulated single-use laparoscopic instrument (ArtiSentialTM). The two patients underwent different laparoscopic interventions at Ukrb University (Neuruppin, Germany): nephrectomy and Anderson-Hynes-pyeloplasty. All procedures were completed, with no need for conversion or placement of additional ports. No intraoperative complications or technical failure of the instrument was recorded. The mean operative time was 180 min median length of stay was 11.5 d. The instrument could be opened out of the sterile packaging and used at once when it was needed, because it is a single-use instrument. There was real haptical feedback and the costs are minimal compared to robot surgery. The use was straightforward and rapid processes after an intensive training of 4 h in a dry lap. Awaiting future investigations in larger series, this study proves the safety and feasibility of renal surgery with ArtiSentialTM and provides relevant data that may help early adopters of this surgical instrument.

Spectra of seismic force reduction factors of MDOF systems normalized by two characteristic periods

Seismic force reduction factor(SFRF) spectra of shear-type multi-degree-of-freedom(MDOF) structures are investigated. The modified Clough model, capable of considering the strength-degradation/hardening and stiffnessdegradation, is adopted. The SFRF mean spectra using 102 earthquake records on a typical site soil type(type C) are constructed with the period abscissa being divided into three period ranges to maintain the peak features at the two sitespecific characteristic periods. Based on a large number of results, it is found that the peak value of SFRF spectra may also exist for MDOF, induced by large high-mode contributions to elastic base shear, besides the mentioned two peak values. The variations of the stiffness ratio λk and the strength ratio λF of the top to bottom story are both considered. It is found that the SFRFs for λF ≤λk are smaller than those for λF > λk. A SFRF modification factor for MDOF systems is proposed with respect to SDOF. It is found that this factor is significantly affected by the story number and ductility. With a specific λF(= λk0.75), SFRF mean spectra are constructed and simple solutions are presented for MDOF systems. For frames satisfying the strong column/weak beam requirement, an approximate treatment in the MDOF shear-beam model is to assign a post-limit stiffness 15%-35% of the initial stiffness to the hysteretic curve. SFRF spectra for MDOF systems with 0.2 and 0.3 times the post-limit stiffness are remarkably larger than those without post-limit stiffness. Thus, the findings that frames with beam hinges have smaller ductility demand are explained through the large post-limit stiffness.

Dynamics of Sea Ice and Spectrum of Ice Force

Spectrum and self-excite characters are the two significant characters of the dynamics of sea ice. The spectrum character of sea ice is mainly shown by the spectrum of ice force. The spectrum character of the sea ice is its intrinsic attributes. When the spectrum of ice force from the dynamic response of ice and structure interaction are evaluated, the effect of dynamic character of the structure must be eliminated. In this paper, the ice force spectrum at Bohai Bay and Liaodong Bay is evaluated from the displacement and strain responses of a single degree and a multi-degree freedom structure. The evaluated ice force spectrum can be used to define the spectrum character of ice in the analysis of ice induced vibration.

Development of Five-Finger Multi-DoF Myoelectric Hands with a Power Allocation Mechanism

To be used as five-fingered myoelectric hands in daily living, robotic hands must be lightweight with the size of human hands. In addition, they must possess the DoFs （degrees of freedom） and high grip force similar to those of human hands. Balancing these requirements involves a trade-off; ideal robotic hands have yet to sufficiently satisfy both requirements. Herein, a power allocation mechanism is proposed to improve the grip force without increasing the size or weight of robotic hands by using redundant DoFs during pinching motions. Additionally, this mechanism is applied to an actual five-fingered myoelectric hand, which produces seven types of motions necessary for activities of daily living and realizes a -60% improvement in fingertip force, allowing three fingers to pinch objects exceeding 1 kg.

THE REDUCED BASIS TECHNIQUE AS A COARSE SOLVER FOR PARAREAL IN TIME SIMULATIONS

In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial differential equation. The fine solver is based on the finite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. Of[line-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space （typically small）. Parareal in time algorithms based on a multi-grids finite element method and a multi-degrees finite element method are also presented. Some numerical results are reported.