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A Practical Way to Characterize Non-degenerate Cubic BÉZIER Curves


  • Cadi Ayyad University - Ecole Supérieure de Technologie de Safi (High College of Technology of Safi), Morocco


Background: The existing geometric characterization diagrams aren’t so detailed to make easier their implementation neither to allow discriminating different classes of curve shapes. The objective is to bring in some refinements and develop formal ways to implement a hodograph characterization based diagram. Methods: The canonical diagram of Stone and DeRose is used conjointly to the hodograph based diagram of deok-soo to develop a more refined hodograph based diagram. The obtained diagram helps to discriminate between the curves having the same geometric characterization in existing diagrams but with shapes quiet different in term of similarity. Some mathematical concepts like characteristic functions and Boolean calculations are used in building the useful core to implement the refined diagram. Findings: Deok-Soo Kim diagram analyzes the geometric characteristics by determining the location of the origin of coordinate system relatively to the hodograph. In many cases, some particular curves have the same geometric characteristics but they are quite different according to other properties like shape similarity for example. These differences aren’t directly supported in Deok-Soo diagram. So, the Deok-Soo diagram is refined to support more precise geometric characteristics of non-degenerate Bézier curves. In the same diagram, the processing of characteristics in a curve having a loop or bearing an arch with conjugate tangent vectors is not based completely on location of the origin even if the concept is very interesting. The Stone and DeRose’s canonical diagram relating to the location of cusps and nodes is generalized and used to explicitly determine, through algebraic quadratic curves, the location regions of the origin to get curves with conjugate tangent vectors. In addition, some binary characteristic functions and bitwise processing properties are developed to give a practical and provable way to produce algorithms of a geometric characterization tool. Improvements/Applications: This refined hodograph based diagram can be further extended to support the study of degenerate cubic Bézier curves. The diagram can be applied in non-linear geometric transformation jobs.


Binary Characteristic functions, Bitwise Processing, Geometric Characteristics, Location Diagram, Non- Degenerate Bézier Curves.

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  • Stone M, DeRose T. Characterizing cubic Bézier curves.Tech. Rep. EDL-88-8, Xerox Palo Alto Research Center:Palo Alto, Calif; 1988.
  • Stone M, DeRose T. A geometric characterization of parametric cubic curves. Journal ACM Transaction on Graphics. 1989; 8(3):147–63.
  • Kim DS. Hodograph approach to geometric characterization of para-metric cubic curves. Computer-Aided Design.1993; 25(10):644–54.
  • Wang CY. Shape classification of the parametric cubic curve and parametric B-spline cubic curve. ComputerAided Design. 1981; 13(4):199–206.
  • Buchin S, Dingyuan L. An affine invariant and its application in computational geometry. Scientia Sinica. 1983; 24(3):259–67.
  • Knuth DE. The TEXbook computers and typesetting. Vol.A. Addison-Wesley, Reading, Massachusetts, USA; 1984.
  • Lamport L. LATEX: A document preparation system.Addison-Wesley, Reading, Massachusetts, USA; 1986.
  • André J, Vatton I. Typesetting of mathematical symbols taking care of optical scaling. Tech. Rep. 1972, INRIA, Campus universitaire de Beaulieu, 35042 RENNES Cedex France ; 1993,
  • André J, Vatton I. Dynamic optical scaling and variablesized characters. Electronic Publishing. 1994; 7(2):231–50.
  • Lazrek A. Curext, typesetting variable-sized curved symbols.Proceedings of the 14th European TEX Conference, EuroTEX 2003; 2003. p. 47–71.
  • Banouni M, Elyaakoubi M, Lazrek A. Dynamic Arabic mathematical fonts. TeX, XML, and Digital Typography, Berlin Heidelberg; 2004. p. 48–53.
  • Benatia MJE, Elyaakoubi M, Lazrek A. Arabic text justification.Proceedings of the TUG 2006 Conference, TUG; 2006. p. 137–46.
  • Berry DM. Stretching letter and slanted-baseline formatting for Arabic, Hebrew and Persian with dittroff/ fforttid and dynamic postscript fonts. Software: Practice and Experience. 1999; 29(15):1417–57.
  • Elyaakoubi M, Lazrek A. Justify just or just justify. Journal of Electronic Publishing. 2010; 13(1).
  • Bayar A, Khalid S. An optimal way to encode the outlines of variable sized arabic letters in a postscript font. Proceedings of the 16thInternationalConference in Central Europe on Computer Graphics, Visualization and Computer Vision, Full Papers Conference, Vaclav Skala - UNION Agency, Na Mazinách, Plzen Czech Republic; 2008. p. 57–64.
  • Bayar A, Khalid S. Optimization of the curvilinear stretching in a postscript font with quadratic Bézier curves.Proceedings of the International Arab Conference of e-Technology Conference, Arab Open University, Amman –Jordan; 2008. p. 97–104.
  • Bayar A, Khalid S. How a font can respect rules of Arabic calligraphy? The International Arab Journal of e-Technology.2009; 1(1):1–18.
  • Bayar A, Khalid S. Parametric curves variations with respect to a given direction. WSCG Journal. 2010; 18(1–3):33–40.
  • Faux ID, Pratt M. Computational geometry for design and manufacture. Ellis Horwood Ltd.: Campus 400, Maylands Ave. Hemel, Hemstead, Herts, HP2 7EZ UK; 1981.
  • Sederberg TW. Computer aided geometric design.Department of Computer Science, Brigham Young University; 2014.


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