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Cálculo da curva de Bézier em PHP

Este é um código em PHP que contém funções que implementam o cálculo da curva de Bézier (quadrática e cúbica). Os algoritmos estão escritos em uma estrutura simples e semelhante a aplicação pura das respectivas equações, para deixar o código o mais limpo possível, e assim, didático.

As curvas de Bézier, cúbica e quadrática:

Bezier Curves

Referências

  • AHLBERG, J. H.; NILSON, E. N.; WALSH, J. L. The Theory of Splines and Their Applications. Academic Press, 1967.

  • BARNHILL, R. E.; RIESENFELD, R. F. Computer Aided Geometric Design. Academic Press, 1974.

  • BAYDAS, S.; KARAKAS, B. Defining a curve as a Bezier curve. Journal of Taibah University for Science, Vol. 13, No. 1, pp. 522-528, 2019.

  • BERTKA, B. T. An Introduction to Bezier Curves, B-Splines, and Tensor Product Surfaces with History and Applications. University of California Santa Cruz. May 30th, 2008.

  • BISWAS, S.; LOVELL, B. C. Bézier and Splines in Image Processing and Machine Vision. Springer-Verlag London Limited, 2008.

  • BOOR, C. A Practical Guide to Splines, Revised Edition. Applied Mathematical Sciences, Volume 27. Springer-Verlag New York, 2001.

  • CASSELMAN, B. From Bézier to Bernstein. Feature Column, Monthly essays on mathematical topics. American Mathematical Society, November 2008. Disponível em: http://www.ams.org/publicoutreach/feature-column/fcarc-bezier

  • FARIN, G. Curves and Surfaces for CAGD: A Practical Guide, 5th Edition. The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling. Morgan Kaufmann Publishers, 2002.

  • FARIN, G.; HANSFORD, D. Mathematical Principles for Scientific Computing and Visualization. A K Peters, 2008.

  • FARIN, G.; HANSFORD, D. Practical Linear Algebra: A Geometry Toolbox, Third Edition. CRC Press, 2014.

  • FARIN, G.; HOSCHEK, J.; KIM, M.-S. Handbook of Computer Aided Geometric Design. Elsevier Science B.V., 2002.

  • FAROUKI, R. T. The Bernstein polynomial basis: a centennial retrospective. Department of Mechanical and Aerospace Engineering, University of California. March 3, 2012.

  • FAROUKI, R. T.; NEFF, C. A. Analytic properties of plane offset curves. Computer Aided Geometric Design, Vol. 7, pp. 83-99, 1990.

  • GALLIER, J. Curves and Surfaces in Geometric Modeling: Theory and Algorithms. Department of Computer and Information Science, University of Pennsylvania, November 21, 2018. Disponível em: https://www.cis.upenn.edu/~jean/gbooks/geom1.html

  • HUGHES, J. F.; VAN DAM, A.; MCGUIRE, M.; SKLAR, D. F.; FOLEY, J. D.; FEINER, S. K.; AKELEY, K. Computer Graphics: Principles and Practice, Third Edition. Pearson Education, 2014.

  • KAMERMANS, M. A Primer on Bézier Curves. Pomax, 2020. Disponível em: https://pomax.github.io/bezierinfo/

  • PRAUTZSCH, H.; BOEHM, W.; PALUSZNY, M. Bezier and B-Spline Techniques. Mathematics and Visualization Series. Springer-Verlag Berlin Heidelberg, 2002.

  • SIMONI, R. Teoria Local das Curvas. Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Curso de Licenciatura em Matemática. Florianópolis, 2005.

  • WEISSTEIN, E. W. Bézier Curve. MathWorld: A Wolfram Web Resource. Disponível em: https://mathworld.wolfram.com/BezierCurve.html