Let S be a closed orientable surface of genus g ≥ 2,and C（S）the curve complex of S.In the paper,we introduce the concepts of 2-path between edges in C（S）,which can be regarded as an analogue to the edge path betw...Let S be a closed orientable surface of genus g ≥ 2,and C（S）the curve complex of S.In the paper,we introduce the concepts of 2-path between edges in C（S）,which can be regarded as an analogue to the edge path between vertices in C（S）.We show that C（S）is 2P-connected,and the 2-diameter of C（S）is infinite.展开更多
A numerical control (NC) tool path of digital CAD model is widely generated as a set of short line segments in machining. However, there are three shortcomings in the linear tool path, such as discontinuities of tange...A numerical control (NC) tool path of digital CAD model is widely generated as a set of short line segments in machining. However, there are three shortcomings in the linear tool path, such as discontinuities of tangency and curvature, huge number of line segments, and short lengths of line segments. These disadvantages hinder the development of high speed machining. To smooth the linear tool path and improve machining efficiency of short line segments, this paper presents an optimal feed interpolator based on G^2 continuous Bézier curves for the linear tool path. First, the areas suitable for fitting are screened out based on the geometric characteristics of continuous short segments (CSSs). CSSs in every area are compressed and fitted into a G^2 Continuous Bézier curve by using the least square method. Then a series of cubic Bézier curves are generated. However, the junction between adjacent Bézier curves is only G^0 continuous. By adjusting the control points and inserting Bézier transition curves between adjacent Bézier curves, the G^2 continuous tool path is constructed. The fitting error is estimated by the second-order Taylor formula. Without iteration, the fitting algorithm can be implemented in real-time environment. Second, the optimal feed interpolator considering the comprehensive constraints (such as the chord error constraint, the maximum normal acceleration, servo capacity of each axis, etc.) is proposed. Simulation and experiment are conducted. The results shows that the proposed method can generate smooth path, decrease the amount of segments and reduce machining time for machining of linear tool path. The proposed research provides an effective method for high-speed machining of complex 2-D/3-D profiles described by short line segments.展开更多
In this paper, we provide a bijection between the set of underdiagonal lattice paths of length n and the set of(2, 2)-Motzkin paths of length n. Besides, we generalize the bijection of Shapiro and Wang(Shapiro L W, Wa...In this paper, we provide a bijection between the set of underdiagonal lattice paths of length n and the set of(2, 2)-Motzkin paths of length n. Besides, we generalize the bijection of Shapiro and Wang(Shapiro L W, Wang C J. A bijection between 3-Motzkin paths and Schr¨oder paths with no peak at odd height. J. Integer Seq., 2009, 12: Article 09.3.2.) to a bijection between k-Motzkin paths and(k-2)-Schr¨oder paths with no horizontal step at even height. It is interesting that the second bijection is a generalization of the well-known bijection between Dyck paths and 2-Motzkin paths.展开更多
基金Supported by the National Natural Science Foundation of China (Grant No.10931005)
文摘Let S be a closed orientable surface of genus g ≥ 2,and C（S）the curve complex of S.In the paper,we introduce the concepts of 2-path between edges in C（S）,which can be regarded as an analogue to the edge path between vertices in C（S）.We show that C（S）is 2P-connected,and the 2-diameter of C（S）is infinite.
基金Supported by National Natural Science Foundation of China(Grant No.50875171)National Hi-tech Research and Development Program of China(863 Program,Grant No.2009AA04Z150)
文摘A numerical control (NC) tool path of digital CAD model is widely generated as a set of short line segments in machining. However, there are three shortcomings in the linear tool path, such as discontinuities of tangency and curvature, huge number of line segments, and short lengths of line segments. These disadvantages hinder the development of high speed machining. To smooth the linear tool path and improve machining efficiency of short line segments, this paper presents an optimal feed interpolator based on G^2 continuous Bézier curves for the linear tool path. First, the areas suitable for fitting are screened out based on the geometric characteristics of continuous short segments (CSSs). CSSs in every area are compressed and fitted into a G^2 Continuous Bézier curve by using the least square method. Then a series of cubic Bézier curves are generated. However, the junction between adjacent Bézier curves is only G^0 continuous. By adjusting the control points and inserting Bézier transition curves between adjacent Bézier curves, the G^2 continuous tool path is constructed. The fitting error is estimated by the second-order Taylor formula. Without iteration, the fitting algorithm can be implemented in real-time environment. Second, the optimal feed interpolator considering the comprehensive constraints (such as the chord error constraint, the maximum normal acceleration, servo capacity of each axis, etc.) is proposed. Simulation and experiment are conducted. The results shows that the proposed method can generate smooth path, decrease the amount of segments and reduce machining time for machining of linear tool path. The proposed research provides an effective method for high-speed machining of complex 2-D/3-D profiles described by short line segments.
基金The NSF(11601020,11501014)of ChinaOrganization Department of Beijing Municipal Committee(2013D005003000012)Science and Technology Innovation Platform-Business Project 2017(PXM2017_014213_000022)
文摘In this paper, we provide a bijection between the set of underdiagonal lattice paths of length n and the set of(2, 2)-Motzkin paths of length n. Besides, we generalize the bijection of Shapiro and Wang(Shapiro L W, Wang C J. A bijection between 3-Motzkin paths and Schr¨oder paths with no peak at odd height. J. Integer Seq., 2009, 12: Article 09.3.2.) to a bijection between k-Motzkin paths and(k-2)-Schr¨oder paths with no horizontal step at even height. It is interesting that the second bijection is a generalization of the well-known bijection between Dyck paths and 2-Motzkin paths.
