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Topology during Subdivision of Bezier Curves I: Angular Convergence & Homeomorphism [article]

J. Li, T. J. Peters, J. A. Roulier
2013 arXiv   pre-print
We use it to show homeomorphism between a Bezier curve and its control polygon under subdivision.  ...  For Bezier curves, subdivision algorithms create control polygons as piecewise linear (PL) approximations that converge in terms of Hausdorff distance.  ...  Related Work The Angular Convergence we show here is based on a previous result showing that the discrete derivatives of the control polygons converge exponentially to the derivatives of the Bézier curve  ... 
arXiv:1211.0353v4 fatcat:5i2qmcicrvhqnm3bnzegtobjea

An Architecture For Rapid Stereoscopic Image Generation [article]

Shaun MCCann, Paul Lister
1995 Tenth Eurographics Workshop on Graphics Hardware  
Our architecture performs polygon derivation based on purely incremental operations and is well suited to implementation in hardware.  ...  A cost effective architecture for the rasterisation of stereo­ imagery based on image derivation is presented.  ...  Also, the generation of secondary polygons from primary polygons using pure derivation techniques based on spatial mapping only may produce a variety of artifacts in a derived polygonal image.  ... 
doi:10.2312/eggh/eggh95/057-062 dblp:conf/egh/McCannL95 fatcat:imn3oxxfpbhuhhwbyrsijbveom

Isotopic equivalence by Bézier curve subdivision for application to high performance computing

K.E. Jordan, J. Li, T.J. Peters, J.A. Roulier
2014 Computer Aided Geometric Design  
This is extended to show a subdivided control polygon that is ambient isotopic to B.  ...  Furthermore, we derive closed-form formulas to compute sufficient numbers of subdivision iterations to achieve homeomorphism and ambient isotopy, respectively.  ...  One of the reports was particularly comprehensive, well beyond typical expectations, for which we express additional thanks.  ... 
doi:10.1016/j.cagd.2014.07.002 fatcat:qnm2hyrlbbgehpgapf72jhvkdm

Reachability Analysis of Discrete-Time Systems With Disturbances

S.V. Rakovic, E.C. Kerrigan, D.Q. Mayne, J. Lygeros
2006 IEEE Transactions on Automatic Control  
affine and the constraints are polygonal.  ...  A persistent disturbance, dependent on the current state and input, acts on the system.  ...  Let be a polyhedral cover of the polygon . First, note that (48a) (48b) Since each nonempty set is a polyhedron, it follows that is a polygon with a easily-derived polyhedral cover.  ... 
doi:10.1109/tac.2006.872835 fatcat:dcsr2jjyk5cflcbzbdsht2pxgu

Model reduction of uncertain systems retaining the uncertainty structure

Yuri Dolgin, Ezra Zeheb
2005 Systems & control letters (Print)  
Then, applying a novel approach, the infinity norm of "distance" between two polygons representing the original and the reduced uncertain systems is minimized.  ...  obtained in intervals, such that the complex value sets of the uncertain original and reduced models will be optimally close to each other on the unit circle.  ...  Note that the distance between a point and a polygon (defined to be the geometrical distance to the closest point of the polygon) is a nonlinear function of the coordinates of the point and the vertices  ... 
doi:10.1016/j.sysconle.2004.10.010 fatcat:tajh7hwmbneczjm3edgcdy43re

Parameterization and applications of Catmull–Rom curves

Cem Yuksel, Scott Schaefer, John Keyser
2011 Computer-Aided Design  
Furthermore, we provide a formulation that bounds the distance of the curve to the control polygon and explain how globally intersection-free Catmull-Rom curves can be generated using these properties.  ...  The behavior of Catmull-Rom curves heavily depends on the choice of parameter values at the control points.  ...  Other researchers [5] have also noted that a good interpolatory curve is one that does not deviate far from its control polygon.  ... 
doi:10.1016/j.cad.2010.08.008 fatcat:27dynxjxnzac5fduk4hzxlotva

Distributed algorithm for controlling scale-free polygonal formations [article]

Hector Garcia de Marina and Bayu Jayawardhana and Ming Cao
2017 arXiv   pre-print
This paper presents a distributed algorithm for controlling the deployment of a team of mobile agents in formations whose shapes can be characterized by a broad class of polygons, including regular ones  ...  Furthermore, the scale of the polygon can be controlled by only one pair of neighboring agents.  ...  CONCLUSIONS In this paper we have presented a distributed algorithm for controlling formations of a broad class of polygonal shapes, including regular ones, defined by a daisy chain topology.  ... 
arXiv:1703.07960v1 fatcat:6nid26u3wrdlfnawjjehhjt7k4

Distributed algorithm for controlling scale-free polygonal formations * *The work of Hector Garcia de Marina was supported by Mistrale project, http://mistrale.eu. The work of Cao was supported in part by the European Research Council (ERC-StG-307207) and the Netherlands Organization for Scientific Research (NWO-vidi-14134)

Hector Garcia de Marina, Bayu Jayawardhana, Ming Cao
2017 IFAC-PapersOnLine  
This paper presents a distributed algorithm for controlling the deployment of a team of agents in order to form a broad class of polygons, including regular ones, where each agent occupies a corner of  ...  Furthermore, the scale of the polygon can be controlled by only one pair of neighboring agents.  ...  This approach serves as a starting point for controlling polygons in the plane, and not only collinear configurations.  ... 
doi:10.1016/j.ifacol.2017.08.152 fatcat:7hmqeb2vjzgt7i5h6peasfzkzm

Flip procedure in geometric approximation of multiple-component shapes -- Application to multiple-inclusion detection [article]

Pierre Bonnelie, Loïc Bourdin, Fabien Caubet, Olivier Ruatta
2018 arXiv   pre-print
two procedures called intersecting control polygons detection and flip procedure.  ...  The first one allows to prevent potential collisions between two parts of the boundary of the approximated shape, and the second one permits to change its topology by dividing a one-component shape into  ...  The authors would like to thank the anonymous reviewers for their valuable comments and suggestions in order to improve the quality of the paper.  ... 
arXiv:1803.02843v1 fatcat:j6ajy5rdc5eozpfyop6owwllpa

Flip procedure in geometric approximation of multiple-component shapes – Application to multiple-inclusion detection

Pierre Bonnelie, Loïc Bourdin, Fabien Caubet, Olivier Ruatta
2016 SMAI Journal of Computational Mathematics  
two procedures called intersecting control polygons detection and flip procedure.  ...  The first one allows to prevent potential collisions between two parts of the boundary of the approximated shape, and the second one permits to change its topology by dividing a one-component shape into  ...  The authors would like to thank the anonymous reviewers for their valuable comments and suggestions in order to improve the quality of the paper.  ... 
doi:10.5802/smai-jcm.16 fatcat:cmtrualq3jbndkezm76ulte66u

Computational Topology for Approximations of Knots [article]

J. Li, T. J. Peters, K. E. Jordan
2013 arXiv   pre-print
Sufficient conditions are given regarding: (1) Hausdorff distance, and (2) a sum of total curvature and derivative.  ...  The primary contributions are: (i) a priori bounds on the number of subdivision iterations sufficient to achieve a PL approximation that is ambient isotopic to the original Bezier curve, and (ii) improved  ...  SinceL is defined on a compact domain, it is a homeomorphism. Note that C is simple and open, so C(t) is one-to-one, and it is obviously onto.  ... 
arXiv:1401.3370v1 fatcat:xna3lvtwffbszf353nsrwzz7na

Computational topology for approximations of knots

Ji Li, T. J. Peters, K. E. Jordan
2014 Applied General Topology  
Sufficient conditions are given regarding: (1) Hausdorff distance, and (2) a sum of total curvature and derivative.  ...  The primary contributions are: (i) a priori bounds on the number of subdivision iterations sufficient to achieve a PL approximation that is ambient isotopic to the original Bézier curve, and (ii) improved  ...  Because the discrete derivative of the control polygon converges to the derivative c AGT, UPV, 2014 Appl. Gen.  ... 
doi:10.4995/agt.2014.2281 fatcat:3dfbyum6c5cxzpjnsxvby2rmku

Isotopic Equivalence from Bezier Curve Subdivision [article]

J. Li, T. J .Peters, J. A. Roulier
2013 arXiv   pre-print
We prove that the control polygon of a Bezier curve B becomes homeomorphic and ambient isotopic to B via subdivision, and we provide closed-form formulas to compute the number of iterations to ensure these  ...  We first show that the exterior angles of control polygons converge exponentially to zero under subdivision.  ...  Based on this, we shall construct an ambient isotopy between a Bézier curve and the control polygon in Section 5.  ... 
arXiv:1211.0354v4 fatcat:5bnvrkuv25asnp3tvnwmslf6aa

Bounding the Distance between 2D Parametric B�zier Curves and their Control Polygon

M. I. Karavelas, P. D. Kaklis, K. V. Kostas
2004 Computing  
Employing the techniques presented by Nairn, Peters and Lutterkort in [1], sharp bounds are firstly derived for the distance between a planar parametric Bézier curve and a parameterization of its control  ...  polygon based on the Greville abscissae.  ...  In [1] the authors achieve in deriving sharp and easily computable bounds on the maximum distance between a functional Bézier segment and its control polygon.  ... 
doi:10.1007/s00607-003-0051-1 fatcat:eoknts34jva43g2wpirkayptv4

Bounding the Distance between 2D Parametric Bézier Curves and their Control Polygon [chapter]

M. I. Karavelas, P. D. Kaklis, K. V. Kostas
2004 Geometric Modelling  
Employing the techniques presented by Nairn, Peters and Lutterkort in [1], sharp bounds are firstly derived for the distance between a planar parametric Bézier curve and a parameterization of its control  ...  polygon based on the Greville abscissae.  ...  In [1] the authors achieve in deriving sharp and easily computable bounds on the maximum distance between a functional Bézier segment and its control polygon.  ... 
doi:10.1007/978-3-7091-0587-0_10 fatcat:isgmeh5fxrby7lz2d7tftme6w4
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