14501670
Curve fitting and fairing using conic splines
We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. Given a set of planar points, we first construct a tangent continuous conic spline by interpolating the points with a quadratic Bezier spline curve or fitting the data with a smooth arc spline. The arc spline can be represented as a piecewise quadratic rational Bezier spline curve. For parts of the G 1 conic spline without an inflection, we can obtain a curvature continuous conic spline by adjusting the tangent direction at the joint point and scaling the weights for every two adjacent rational Bezier curves. The unwanted curvature extrema within conic segments or at some joint points can be removed efficiently by scaling the weights of the conic segments or moving the joint points along the normal direction of the curve at the point. In the end, a fair conic spline curve is obtained that is G 2 continuous at convex or concave parts and G 1 continuous at inflection points. The main advantages of the method lies in two aspects, one advantage is that we can construct a curvature continuous conic spline by a local algorithm, the other one is that the curvature plot of the conic spline can be controlled efficiently. The method can be used in the field where fair shape is desired by interpolating or approximating a given point set. Numerical examples from simulated and real data are presented to show the efficiency of the new method. q 2003 Elsevier Ltd. All rights reserved.
18068756
Uniform hyperbolic polynomial B-spline curves
This paper presents a new kind of uniform splines, called hyperbolic polynomial B-splines, generated over the space Ω = span{sinht, cosht, tk-3, tk-4 ...,t, 1} in which k is an arbitrary integer larger than or equal to 3. Hyperbolic polynomial B-splines share most of the properties as those of the B-splines in the polynomial space. We give the subdivision formulae for this new kind of curves and then prove that they have the variation dimishing properties and the control polygons of the subdivisions converge. Hyperbolic polynomial B-splines can take care of freeform curves as well as some remarkable curves such as the hyperbola and the catenary. The generation of tensor product surfaces by these new splines is straightforward. Examples of such tensor product surfaces: the saddle surface, the catenary cylinder, and a certain kind of ruled surface are given in this paper.
2621753
High accuracy approximation of helices by quintic curves
In this paper we present methods for approximating a helix segment by quintic Bezier curves or quintic rational Bezier curves based on the geometric Hermite interpolation technique in space. The fitting curve interpolates the curvatures as well as the Frenet frames of the original helix at both ends. We achieve a high accuracy of the approximation by giving a proper parametrization of the curve, and the approximation order of the height function along the helix axis is 9 provided that the screw angle of the helix is fixed. Numerical examples are also presented to illustrate the efficiency of the new method.
13366253
Efficient circular arc interpolation based on active tolerance control
In this paper, we present an efficient sub-optimal algorithm for fitting smooth planar parametric curves by G1 arc splines. To fit a parametric curve by an arc spline within a prescribed tolerance, we first sample a set of points and tangents on the curve adaptively as well as with enough density, so that an interpolation biarc spline curve can be with any desired high accuracy. Then, we construct new biarc curves interpolating local triarc spirals explicitly based on the control of permitted tolerances. To reduce the segment number of fitting arc spline as much as possible, we replace the corresponding parts of the spline by the new biarc curves and compute active tolerances for new interpolation steps. By applying the local biarc curve interpolation procedure recursively and sequentially, the result circular arcs with no radius extreme are minimax-like approximation to the original curve while the arcs with radius extreme approximate the curve parts with curvature extreme well too, and we obtain a near optimal fitting arc spline in the end. Even more, the fitting arc spline has the same end points and end tangents with the original curve, and the arcs will be jointed smoothly if the original curve is composed of several smooth connected pieces. The algorithm is easy to be implemented and generally applicable to circular arc interpolation problem of all kinds of smooth parametric curves. The method can be used in wide fields such as geometric modeling, tool path generation for NC machining and robot path planning, etc. Several numerical examples are given to show the effectiveness and efficiency of the method.
3643118
Skeleton-driven 2D distance field metamorphosis using intrinsic shape parameters
In this article a novel algorithm is presented for 2D shape interpolation using the intrinsic shape parameters of a piecewise linear curve. The skeletons of two given shapes are computed and the smooth transformation of distance fields is driven by metamorphosis from the skeleton of the source object to that of the target one. We introduce feature graphs, linear forms of skeletons, to guide the construction of intermediate skeleton. If the topologies of the source object and the target one are different, their feature graphs will be automatically extended with equivalent topologies. Then we apply the technique of intrinsic shape parameters to the smooth transition of the extended feature graphs, which will guide the metamorphosis of the skeletons. Not only can the new approach be capable of morphing between objects with different topological genus, but also the topologies and the shapes of the intermediate objects can be controlled efficiently.
15110446
Normal based subdivision scheme for curve design
In this paper we propose a new kind of nonlinear and geometry driven subdivision scheme for curve interpolation. Instead of using linear combination of old vertexes, displacement vector for every new vertex is given by normal vectors at old vertexes. The normal vectors are computed adaptively for each time of subdivision, and the limit curve is G^1 smooth with wide ranges of free parameters. With this new scheme, normal vectors at selected vertexes can be interpolated efficiently. A shape preserving subdivision scheme with explicit choices of all free parameters is also presented.
18115243
A wireless MAC protocol using implicit pipelining
In distributed multiple access control protocols, two categories of overhead are usually associated with contention resolution. One is channel idle overhead, where all contending stations are waiting to transmit. Another is collision overhead, which occurs when multiple contending stations attempt to transmit simultaneously. Either idle overhead or collision overhead being large, contention resolution algorithm would be inefficient. Prior research work tries to minimize both the idle and the collision overheads using various methods. In this paper, we propose to apply "pipelining" techniques to the design of multiple access control protocol so that channel idle overhead could be (partially) hidden and the collision overhead could be reduced. While the concept of pipelined scheduling can be applied to various MAC protocol designs in general, in this paper, we focus on its application to IEEE 802.11 DCF. In particular, an implicitly pipelined dual-stage contention resolution MAC protocol (named DSCR) is proposed. With IEEE 802.11, the efficiency of contention resolution degrades dramatically with the increasing load due to high probability of collision. Using the implicit pipelining technique, DSCR hides the majority of channel idle time and reduces the collision probability, hence, improves channel utilization, average access delay, and access energy cost over 802.11 significantly both in wireless LANs and in multihop networks. The simulation results, as well as some analysis, are presented to demonstrate the effectiveness of DSCR.
1833949
DSCR: a wireless MAC protocol using implicit pipelining
In wireless networks, the collision cost is much higher than wired networks since a station cannot detect collision until a transmission is over and the expected acknowledgment does not come back. Therefore, more efficient contention resolution algorithms are desired for wireless networks to reduce the collision probability among contending stations. IEEE 802.11 defines a distributed random access MAC protocol named DCF (Distributed Coordination Function), which uses binary exponential backoff (BEB) algorithm to resolve channel contention. In DCF, a station wanting to access the channel generates a random backoff counter uniformly distributed over the interval [0, CW] (CW represents contention window). This backoff counter corresponds to the number of idle slots this station has to wait before its transmission. The contention window, CW, is exponentially increased by a factor of 2 each time when a collision happens, and reset to the minimum value upon successful transmission. An appropriate choice of CW can optimize the performance of 802.11. However, the optimum value of CW changes with the network size and 802.11 operates far from this optimum point. In particular, in a heavily contended network, the collision probability increases significantly, degrading the performance of 802.11. To improve the performance of IEEE 802.11, [1] proposes that each station continuously observes channel activities and estimates the number of contending stations, hence, dynamically tune the value of CW to the optimum point. [2] also proposes a fast collision resolution algorithm, in which the winning station occupies the channel for a certain period of time to reduce the collision probability. However, both of these two schemes are designed for wireless LANs, where all stations are within each other’s transmission range. This paper proposes a MAC protocol, named DSCR (Dual Stage Contention Resolution), uses “pipelined” two stage contention resolution algorithm to reduce collision probability and achieve better channel utilization than 802.11 in both wireless LANs and ad hoc networks. In our prior work, we developed two explicit pipelining schemes, which use a separate control channel (either to transmit RTS/CTS or to send busy tone) to pipeline contention resolution with data transmission [4]. In contrast, DSCR performs implicit pipelining in that no separate control channel is required.
34097947
Multiobjective second-order symmetric duality with F
We suggest a pair of second-order symmetric dual programs in multiobjective nonlinear programming. For these second-order symmetric dual programs, we prove the weak, strong and converse duality theorems under F-convexity conditions.
40840411
A property on convex fuzzy sets
In this note, a new result characterizing fuzzy set convexity is given under lower semicontinuous condition.
12349846
Second order symmetric duality in non-differentiable multiobjective programming with F
Abstract This paper is concerned with a pair of Mond–Weir type second order symmetric dual non-differentiable multiobjective programming problems. We establish the weak and strong duality theorems for the new pair of dual models under second order F -convexity assumptions. Several results including many recent works are obtained as special cases.