Efficient Construction of B-Spline Curves with Minimal Internal Energy

In this paper,we propose an efficient method to construct energy-minimizing B-spline curves by using discrete mask method.The linear relations between control points are firstly derived for different energy-minimization problems,then the construction of B-spline curve with minimal internal energy can be addressed by solving a sparse linear system.The existence and uniqueness of the solution for the linear system are also proved.Experimental results show the efficiency of the proposed approach,and its application in 1 G blending curve construction is also presented.

A Novel Contour Tracing Algorithm for Object Shape Reconstruction Using Parametric Curves

Parametric curves such as Bézier and B-splines, originally developedfor the design of automobile bodies, are now also used in image processing andcomputer vision. For example, reconstructing an object shape in an image,including different translations, scales, and orientations, can be performedusing these parametric curves. For this, Bézier and B-spline curves can be generatedusing a point set that belongs to the outer boundary of the object. Theresulting object shape can be used in computer vision fields, such as searchingand segmentation methods and training machine learning algorithms. Theprerequisite for reconstructing the shape with parametric curves is to obtainsequentially the points in the point set. In this study, a novel algorithm hasbeen developed that sequentially obtains the pixel locations constituting theouter boundary of the object. The proposed algorithm, unlike the methods inthe literature, is implemented using a filter containing weights and an outercircle surrounding the object. In a binary format image, the starting point ofthe tracing is determined using the outer circle, and the next tracing movementand the pixel to be labeled as the boundary point is found by the filter weights.Then, control points that define the curve shape are selected by reducing thenumber of sequential points. Thus, the Bézier and B-spline curve equationsdescribing the shape are obtained using these points. In addition, differenttranslations, scales, and rotations of the object shape are easily provided bychanging the positions of the control points. It has also been shown that themissing part of the object can be completed thanks to the parametric curves.

Approximate merging of B-spline curves and surfaces

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.

One Fairing Method of Cubic B-spline Curves Based on Weighted Progressive Iterative Approximation

A new method to the problem of fairing planar cubic B-spline curves is introduced in this paper. The method is based on weighted progressive iterative approximation （WPIA for short） and consists of following steps： finding the bad point which needs to fair, deleting the bad point, re-inserting a new data point to keep the structm-e of the curve and applying WPIA method with the new set of the data points to obtain the faired curve. The new set of the data points is formed by the rest of the original data points and the new inserted point. The method can be used for shape design and data processing. Numerical examples are provided to demonstrate the effectiveness of the method.

The RXTE/PCA 35-Day X-Ray Lightcurve of Hercules X-1

The archival Rossi X-ray Timing Explorer (RXTE) Proportional Counter Array (PCA) observations of HZ Her/Her X-1 are analyzed here. The time periods for Anomalous Low States are refined. The 35-day X-ray lightcurve of Her X-1 is produced using 35-day phases determined with RXTE All-Sky Monitor (ASM) and Swift Burst Alert Telescope (BAT) data, with adjustments based on the RXTE/PCA data. As a result, we constructed the best-phased 35-day lightcurve of Her X-1 with the highest sensitivity. The RXTE/PCA data are separated into 8 different substates of the 35-day cycle: Main High (MH) turn-on, MH, MH decline, Low State (LS) 1, Short High (SH) turn-on, SH, SH decline, and LS2. Orbital lightcurves are created for each substate, and they are compared. Eclipse data are extracted for each substate to determine the changes in eclipse shape with 35-day phase. A scattering corona model is fit to the eclipses for the different 35-day substates: this shows that the corona is present with similar intensity at all 35-day phases. We search for previously reported dips without absorption but find none.

RECONSTRUCTION OF SYMMETRIC B-SPLINE CURVES AND SURFACES

A method to reconstruct symmetric B-spline curves and surfaces is presented. The symmetry property is realized by using symmetric knot vector and symmetric control points. Firstly, data points are divided into two parts based on the symmetry axis or symmetry plane extracted from data points. Then the divided data points are parameterized and a symmetric knot vector is selected in order to get symmetric B-spline basis functions. Constraint equations regarding the control points are deduced to keep the control points of the B-spline curve or surface to be symmetric with respect to the extracted symmetry axis or symmetry plane. Lastly, the constrained least squares fitting problem is solved with the Lagrange multiplier method. Two examples from industry are given to show that the proposed method is efficient, robust and able to meet the general engineering requirements.

THE RELATIONSHIP BETWEEN PROJECTIVE GEOMETRIC AND RATIONAL QUADRATIC B-SPLINE CURVES

Abstract For two rational quadratic B spline curves with same control vertexes, the cross ratio of four collinear points are represented: which are any one of the vertexes, and the two points that the ray initialing from the vertex intersects with the corresponding segments of the two curves, and the point the ray intersecting with the connecting line between the two neighboring vertexes. Different from rational quadratic Bézier curves, the value is generally related with the location of the ray, and the necessary and sufficient condition of the ratio being independent of the ray's location is showed. Also another cross ratio of the following four collinear points are suggested, i.e. one vertex, the points that the ray from the initial vertex intersects respectively with the curve segment, the line connecting the segments end points, and the line connecting the two neighboring vertexes. This cross ratio is concerned only with the ray's location, but not with the weights of the curve. Furthermore, the cross ratio is projective invariant under the projective transformation between the two segments.

Spanning Pre-disks in a Compression Body

A properly embedded essential planar surface P（not a disk） in a compression body V is called a spanning pre-disk with respect to J, if one boundary component of P is lying in ?;V and all other boundary components of P are lying in?;V and coplanar with J. In this paper, we show that the number of boundary components of spanning pre-disks in a compression body is unbounded. But the number of a maximal collection of spanning pre-disks is bounded.

AHT Bézier Curves and NUAHT B-Spline Curves

In this paper, we present two new unified mathematics models of conics and polynomial curves, called algebraic hyperbolic trigonometric （ AHT） Bezier curves and non-uniform algebraic hyperbolic trigonometric （ NUAHT） B-spline curves of order n, which are generated over the space span{sin t, cos t, sinh t, cosh t, 1, t,..., t^n-5}, n 7〉 5. The two kinds of curves share most of the properties as those of the Bezier curves and B-spline curves in polynomial space. In particular, they can represent exactly some remarkable transcendental curves such as the helix, the cycloid and the catenary. The subdivision formulae of these new kinds of curves are also given. The generations of the tensor product surfaces are straightforward. Using the new mathematics models, we present the control mesh representations of two classes of minimal surfaces.

Constructing iterative non-uniform B-spline curve and surface to fit data points

In this paper, based on the idea of profit and loss modification, we presentthe iterative non-uniform B-spline curve and surface to settle a key problem in computeraided geometric design and reverse engineering, that is, constructing the curve (surface)fitting (interpolating) a given ordered point set without solving a linear system. We startwith a piece of initial non-uniform B-spline curve (surface) which takes the given point setas its control point set. Then by adjusting its control points gradually with iterative formula,we can get a group of non-uniform B-spline curves (surfaces) with gradually higherprecision. In this paper, using modern matrix theory, we strictly prove that the limit curve(surface) of the iteration interpolates the given point set. The non-uniform B-spline curves(surfaces) generated with the iteration have many advantages, such as satisfying theNURBS standard, having explicit expression, gaining locality, and convexity preserving,etc

G2连续条件下圆域B样条曲线的延拓与拼接

为克服圆域B样条曲线在C2连续条件下延拓到目标圆盘时的形状不可调整性,提出利用圆域Bézier曲线在G2连续条件下延拓圆域B样条曲线的算法。首先通过极小化能量目标函数确定延拓部分自由度,使延拓后的圆域曲线形状达到最优状态。然后将节点向量重新参数化,令得到的混合圆域曲线转化为圆域B样条曲线,重复利用该算法可以把圆域B样条曲线延拓至多个目标圆盘。此外提出利用六次圆域Bézier曲线作为过渡曲线拼接任意两条圆域B样条曲线的算法,最后给定数值实例验证算法的普适性和有效性。

Generation of Discrete Bicubic G^1 B-Spline Ship Hullform Surfaces from a Given Curve Notwork Using Virtual Iso-Parametric Curves

We propose a method that automatically generates discrete bicubic G^1 continuous B-spline surfaces that interpolate the curve network of a ship huliform.First,the curves in the network are classified into two types;boundary curves and ＂reference curves＂,The boundary curves correspond to a set of rectangular（or triangular）topological type that can be representes with tensot-product （or degenerate）B-spline surface patches.Next,in the interior of the patches,surface fitting points and cross boundary derivatives are estimated from the reference curves by constructing ＂virtual＂isoparametric curves.Finally,a discrete G^1 continuous B-spline surface is gencrated by a surface fitting algorithm.Several smooth ship hullform surfaces generated from curve networks corresponding to actual ship hullforms demonstrate the quality of the method.

A matrix method for degree-raising of B-spline curves

A new identity is proved that represents the kth order B-splines as linear combinations of the (k + 1) th order B-splines A new method for degree-raising of B-spline curves is presented based on the identity. The new method can be used for all kinds of B-spline curves, that is, both uniform and arbitrarily nonuniform B-spline curves. When used for degree-raising of a segment of a uniform B-spline curve of degree k - 1, it can help obtain a segment of curve of degree k that is still a uniform B-spline curve without raising the multiplicity of any knot. The method for degree-raising of Bezier curves can be regarded as the special case of the new method presented. Moreover, the conventional theory for degree-raising, whose shortcoming has been found, is discussed.

Visual Modeling of Rice Root Growth Based on B-Spline Curve

As a major food production crop in China,the growth and development of rice is an extremely complex systemic process,and the root system is the main organ for rice to obtain nutrients.Therefore,3D modeling and visualization of the rice root system can help to further understand its morphology,structure and function,and provide an aid for scientific cultivation of rice and improving rice yield for decision making.In this paper,a mathematical model of the rice root system is established based on the B spline curve combined with the L-system approach,using mathematical knowledge based on the 3D morphological characteristics of the real rice root system.The B-Spline Curve is chosen to simulate this,and the recursive definition of B-Spline Curve and its formula are used to realize the modeling of the rice root system curve.Based on the mathematical method of rice root system integration,the bending effect of rice root system at different periods and different growth positions is realized.Finally,the L-system combined with B-Spline Curve is used to construct a rice root system model and realize the rice root system visualization simulation.The simulated image is closer to the real rice root system image in terms of morphological structure and has a strong sense of realism.

KNOT PLACEMENT FOR B-SPLINE CURVE APPROXIMATION VIA l_(∞,1)-NORM AND DIFFERENTIAL EVOLUTION ALGORITHM

In this paper,we consider the knot placement problem in B-spline curve approximation.A novel two-stage framework is proposed for addressing this problem.In the first step,the l_(∞,1)-norm model is introduced for the sparse selection of candidate knots from an initial knot vector.By this step,the knot number is determined.In the second step,knot positions are formulated into a nonlinear optimization problem and optimized by a global optimization algorithm—the differential evolution algorithm(DE).The candidate knots selected in the first step are served for initial values of the DE algorithm.Since the candidate knots provide a good guess of knot positions,the DE algorithm can quickly converge.One advantage of the proposed algorithm is that the knot number and knot positions are determined automatically.Compared with the current existing algorithms,the proposed algorithm finds approximations with smaller fitting error when the knot number is fixed in advance.Furthermore,the proposed algorithm is robust to noisy data and can handle with few data points.We illustrate with some examples and applications.

The Location of Singular and Inflection Points for Planar Cubic B-Spline Curve

Using vectors between control points(a_i=P_(i+1)-P_i),parameters λ and μ(such that a_(i+1)=λ_(ai+μ_(a_i+2))are used to study the shape classification of planar parametric cubic B-spline curves. The regiosn of λμ space corresponding to different geometric features on the curves are investigated.These results are useful for curve design.

密植果园开沟施肥机开沟刀片设计与试验

针对现有开沟刀片在密植果园开沟施肥过程中存在功率消耗大、沟深稳定性差的问题,运用旋耕理论和试验分析的方法,设计了一种正弦指数曲线型开沟刀片。通过分析拖拉机前进速度、刀盘转速以及二者相互作用对功率消耗和沟深稳定性的影响,确定了正弦指数曲线型开沟刀片工作参数的最佳组合为:拖拉机前进速度1 m/s,刀盘转速为200 r/min。以正弦指数曲线型开沟刀片B处弯曲角、弯曲半径、C处弯曲角为影响因子,以功率消耗和沟深稳定性为响应值,通过三因素四水平正交试验,确定了正弦指数曲线型开沟刀片结构参数的最佳组合为:B处弯曲角为95°,C处弯曲角为140°,弯曲半径为9 mm。对正弦指数曲线型开沟刀片和现有开沟刀片进行对比试验,结果表明:在工作参数和开沟深度相同的条件下,正弦指数曲线型开沟刀片比现有开沟刀片功率消耗下降3.29 k W,沟深稳定性提高8.83个百分点,各项技术指标均达到了国家相关标准的技术要求。

2BFD-2C型花生覆膜播种机的设计

针对我国北方垄作花生种植模式的特点,开发设计了2BFD-2C型花生覆膜播种机。该播种机主要由机架、悬挂装置、排肥系统、排种系统、起垄装置、传动系统、喷药系统、覆膜装置及膜上覆土装置等组成,一次完成施肥、播种、起垄、覆膜、膜上覆土、镇压以及喷洒除草剂等作业,机具结构精简化设计,大大降低了制造成本。田间试验结果表明:播种机双穴率达到96%,作业质量符合花生种植的农艺要求,为我国垄作花生种植提供了技术支持。