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Monotone Arc Diagrams with few Biarcs [article]

Steven Chaplick, Henry Förster, Michael Hoffmann, Michael Kaufmann
2020 arXiv   pre-print
We show that every planar graph can be represented by a monotone topological 2-page book embedding where at most 15n/16 (of potentially 3n-6) edges cross the spine exactly once.  ...  Figure 1 1 Arc diagram (a), monotone arc diagram (b), proper arc diagram (c) of the octahedron. . As a main result, we improve the upper bound on the number of monotone biarcs: Theorem 1.1.  ...  [7] used monotone arc diagrams with only downup biarcs to construct small universal point sets for 1-bend drawings of planar graphs.  ... 
arXiv:2003.05332v1 fatcat:qdaiwf4mffas7akhn63ejd2hrq

Numerical methods for approximating digitized curves by piecewise circular arcs

Shi-Nine Yang, Wei-Chang Du
1996 Journal of Computational and Applied Mathematics  
First, iterative methods are proposed to solve the best single arc and biarc approximation problems of a digitized curve with respect to the maximum norm.  ...  In this paper, we propose new algorithms to approximate digitized curves by piecewise circular arcs with geometric continuity G O or G t.  ...  the digitized curve, and (3) the number of arcs is as few as possible.  ... 
doi:10.1016/0377-0427(95)00191-3 fatcat:fsq55elfyfhuzjj6mjftctasj4

Minkowski sum computation for planar freeform geometric models using $$G^1$$ G 1 -biarc approximation and interior disk culling

Sangjun Han, Seung-Hyun Yoon, Myung-Soo Kim, Gershon Elber
2019 The Visual Computer  
The boundary curves are first approximated by G 1 -biarc splines within a given error bound > 0. A superset of Minkowski sum boundary is then generated using the biarc approximations.  ...  From the planar arrangement of remaining arcs, we construct the Minkowski sum boundary in a correct topology.  ...  [39] , the planar curves are first subdivided into spiral curves (with monotone curvature), for a better performance of G 1 -biarc approximation.  ... 
doi:10.1007/s00371-019-01687-6 fatcat:vakynai4o5h7zbipcparlsjwv4

A positive fraction Erdos-Szekeres theorem and its applications [article]

Andrew Suk, Ji Zeng
2022 arXiv   pre-print
We apply our results to mutually avoiding planar point sets and biarc diagrams in graph drawing.  ...  For n > (k-1)^2, any sequence A of n distinct real numbers contains a collection of subsets A_1,..., A_k ⊂ A, appearing sequentially, all of size s=Ω(n/k^2), such that every subsequence (a_1,..., a_k), with  ...  Hence, we can decompose Figure 6 : 6 Figure 6: (i) a proper arc diagram. (ii) a monotone biarc diagram. Block-monotone sequence partitionThis section is devoted to the proof of Theorem 1.2.  ... 
arXiv:2112.01750v2 fatcat:wtu642vel5hh7gzkpklugof4h4

COMPUTATIONAL AND STRUCTURAL ADVANTAGES OF CIRCULAR BOUNDARY REPRESENTATION

OSWIN AICHHOLZER, FRANZ AURENHAMMER, THOMAS HACKL, BERT JÜTTLER, MARGOT RABL, ZBYNEK ŠÍR
2011 International journal of computational geometry and applications  
Boundary approximation of planar shapes by circular arcs has quantitative and qualitative advantages compared to using straight-line segments.  ...  Fig. 6 : 6 Small point sample (dashed Voronoi diagram) versus few arcs (solid medial axis) A and a point p on its medial axis M (A), denote with D p the unique maximal disk with center p.  ...  When starting the next biarc from y with r y = 1/k y (unless y is an apex), monotonicity of signed curvature will be preserved.  ... 
doi:10.1142/s0218195911003548 fatcat:qhulxzkzpzbwhmc3lay4rwbve4

Various Types of Aesthetic Curves [article]

R.U. Gobithaasan
2013 arXiv   pre-print
Farin defined a fair curve as a curve which generates continuous curvature profile and consists of only a few monotonic pieces (pg.364, [14] ).  ...  In 1990s, biarcs are available in NC machines thus many algorithms have been developed for curve fitting with biarcs, e.g., Schonherr [45] developed a biarc curve fitting technique which minimizes the  ... 
arXiv:1304.7881v1 fatcat:ui3h3dulvbe5fctseioipzbtca

Coaxing a planar curve to comply

D.S. Meek
2002 Journal of Computational and Applied Mathematics  
We illustrate almost all the methods discussed with diagrams.  ...  We start with methods invented by Newton (1643-1727) and Lagrange (1736 -1813) and proceed to recent methods that are the subject of current research.  ...  The biarc is useful in situations where circular arcs are desirable curves, such as in machining where a circular arc is a natural and easy-to-cut curve.  ... 
doi:10.1016/s0377-0427(01)00478-2 fatcat:jr43wuflt5ftborljhiyfdeoum

Exact Medial Axis Computation for Circular Arc Boundaries [chapter]

Oswin Aichholzer, Wolfgang Aigner, Thomas Hackl, Nicola Wolpert
2012 Lecture Notes in Computer Science  
However, we show how to avoid inaccuracies in the medial axis computations arising from a non-algebraic biarc construction of the boundary.  ...  We finally show that all necessary computations can be performed over the field of rational numbers with a small number of adjoint square-roots.  ...  But with the help of the properties of the ECAB structure (as opposed to numerical biarc constructions), and by modifying a few specific steps, a mathematically correct representation of the medial axis  ... 
doi:10.1007/978-3-642-27413-8_2 fatcat:6annfpv7xnf2thrj6meyfdqpsm

Generating parametric models of tubes from laser scans

Ulrich Bauer, Konrad Polthier
2009 Computer-Aided Design  
Our main contributions are reconstruction of the spine curve of a pipe surface from surface samples, and approximation of the spine curve by G 1 continuous circular arcs and line segments.  ...  Biarcs, first introduced in [21] , are easier to handle than regular arcs, because both endpoint tangents of a biarc can be chosen independently.  ...  In the general case, a biarc with fixed end points and tangents has only one degree of freedom for choosing the junction point. Now consider a biarc with coinciding start and end points and tangents.  ... 
doi:10.1016/j.cad.2009.01.002 fatcat:4bqw72mi4jaavagnghhexecosy

Parametric Reconstruction of Bent Tube Surfaces

Ulrich Bauer, Konrad Polthier
2007 2007 International Conference on Cyberworlds (CW'07)  
Our main contributions are reconstruction of the spine curve of a pipe surface from surface samples, and approximation of the spine curve by G 1 continuous circular arcs and line segments.  ...  biarcs.  ...  Biarcs, first introduced in [19] , are easier to handle than regular arcs, because both endpoint tangents of a biarc can be chosen independently.  ... 
doi:10.1109/cw.2007.59 dblp:conf/cw/BauerP07 fatcat:lw324gwsczbfncg5adhte6xjt4

An arc spline approximation to a clothoid

D.S. Meek, D.J. Walton
2004 Journal of Computational and Applied Mathematics  
It is proved that if the arc spline has n arcs, then the error in the approximation is of order O(1=n 2 ).  ...  Here the clothoid is approximated by an arc spline. The chief advantage in doing so is that arc splines are very easy to lay out and to o set.  ...  For example, Hickerson used biarcs, triarcs [5, p. 132] , and quadarcs [5, p. 139 ], but did not arrange that the curvatures vary linearly with arc length.  ... 
doi:10.1016/j.cam.2003.12.038 fatcat:z4xkjinvxvaojdhuku55iybog4

A Review on Approaches for Handling Bezier Curves in CAD for Manufacturing

Hetal N. Fitter, Akash B. Pandey, Divyang D. Patel, Jitendra M. Mistry
2014 Procedia Engineering  
Various techniques and methodologies like curve fitting, curve manipulation, blending and merging of curves have been proposed over the years for better handling and enhancing Bezier curve use with every  ...  Quadratic Bezier curves are used very well as transition curves, can substitute biarcs (two circular arcs) owing to polynomial advantage, used to approximate circular arcs, its computation cost is less  ...  Assuming a planar Bezier curve with smooth and monotonous curvature variation in mind, fixing degrees to be chosen in the range of 3 to 5, he proposed a curve fitting method with shape characterization  ... 
doi:10.1016/j.proeng.2014.12.394 fatcat:dqa53x5ijnaq5nczua5vybi7su

Computational and Structural Advantages of Circular Boundary Representation [chapter]

Oswin Aichholzer, Franz Aurenhammer, Thomas Hackl, Bert Jüttler, Margot Oberneder, Zbyněk Šír
Lecture Notes in Computer Science  
Boundary approximation of planar shapes by circular arcs has quantitative and qualitative advantages compared to using straight-line segments.  ...  Fig. 6 : 6 Small point sample (dashed Voronoi diagram) versus few arcs (solid medial axis) A and a point p on its medial axis M (A), denote with D p the unique maximal disk with center p.  ...  When starting the next biarc from y with r y = 1/k y (unless y is an apex), monotonicity of signed curvature will be preserved.  ... 
doi:10.1007/978-3-540-73951-7_33 fatcat:aivmixxkrbfptnaeexzvk5itd4

Unported license Drawing Graphs and Maps with Curves

Stephen Kobourov, Martin Nöllenburg, Monique Teillaud, Stephen Kobourov, Martin Nöllenburg, Monique Teillaud, Stephen Kobourov, Martin Nöllenburg, Monique Teillaud, Dagstuhl Reports
Dagstuhl Reports   unpublished
Finally, the seminar was accompanied by the art exhibition Bending Reality: Where Arc and Science Meet with 40 exhibits contributed by the seminar participants.  ...  This report documents the program and the outcomes of Dagstuhl Seminar 13151 "Drawing Graphs and Maps with Curves".  ...  -Drawing Graphs and Maps with  ... 
fatcat:lmmkyc56hre63k45wouhqi7kgq

Ideal knots and other packing problems of tubes

Henryk Gerlach
2009
A closed curve with positive thickness has a self-avoiding neighbourhood that consists of a disjoint union of normal disks with radius ∆, which is a tube. The thesis has three main parts.  ...  We show that thickness defined by global radius of curvature coincides with the notion of thickness based on normal injectivity radius in S 3 .  ...  arc-length parametrised biarc curve.  ... 
doi:10.5075/epfl-thesis-4601 fatcat:yq3s343ldbgddk3p3jg6orhuvy
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