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Bicubic parametric patches are widely used in various geometric applications. These patches are critical in CAD/CAM systems, which are applied in the automotive industry and mechanical and civil engineering. Commonly, Hermite, Bézier, Coons, or NURBS patches are employed in practice. However, the construction of the Hermite bicubic patch is often not easy to explain formally. This contribution presents a new formal method for constructing the Hermite bicubic plate based on the tensor product approach.
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V. Anand. Computer Graphics and Geometric Modeling for Engineers. 1st edn. John Wiley & Sons, Inc., USA, 1993.
P. Bézier. The Mathematical Basis of the UNIURF CAD System. Butterworth-Heinemann, 1986. https://doi.org/10.1016/C2013-0-01005-5. (Crossref)
E. Cohen, R. F. Riesenfeld, and G. Elber. Geometric Modeling with Splines: An Introduction. A K Peters/CRC Press, 2019. https://doi.org/10.1201/9781439864203. (Crossref)
G. Farin. Bézier triangles. In: G. Farin (Ed.), Curves and Surfaces for Computer-Aided Geometric Design, 3rd edn., chap. 18, pp. 321-351. Academic Press, Boston, 1993. https://doi.org/10.1016/B978-0-12-249052-1.50023-4. (Crossref)
R. Goldman. An Integrated Introduction to Computer Graphics and Geometric Modeling. 1st edn. CRC Press, Inc., USA, 2009. https://doi.org/10.1201/9781439803356. (Crossref)
D. J. Holliday and G. E. Farin. Geometric interpretation of the diagonal of a tensor-product Bézier volume. Computer Aided Geometric Design 16(8):837-840, 1999. https://doi.org/10.1016/S0167-8396(99)00004-7. (Crossref)
A. Kolcun. Biquadratic S-Patch in Bézier form. In: WSCG 2011 Communication Papers Proceedings - Proc. 19th International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision, vol. 19, pp. 201-207, 2011. http://wscg.zcu.cz/WSCG2011/!_2011_WSCG-Short_Papers.pdf.
N. Mochizuki. The tensor product of function algebras. Tohoku Mathematical Journal 17(2):139-146, 1965. https://doi.org/10.2748/tmj/1178243579. (Crossref)
H. Prautzsch and W. Boehm. Geometric Concepts for Geometric Design. A K Peters/CRC Press, 1993. https://doi.org/10.1201/9781315275475. (Crossref)
A. Rockwood and P. Chambers. Interactive Curves and Surfaces: A Multimedia Tutorial on CAGD. 1st edn. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1996.
V. Skala. New geometric continuity solution of parametric surfaces. AIP Conference Proceedings 1558:2500-2503, 2013. https://doi.org/10.1063/1.4826048. (Crossref)
V. Skala. Hermite parametric bicubic patch defined by the tensor product. In: Computational Science and Its Applications - ICCSA 2022, vol. 13376 of Lecture Notes in Computer Science, pp. 228-235. Springer, 2022. https://doi.org/10.1007/978-3-031-10450-3_18. (Crossref)
V. Skala and V. Ondracka. BS-Patch: Constrained Bézier parametric patch. WSEAS Transactions on Mathematics 12(5):598-607, 2013. https://wseas.com/journals/articles.php?id=5799.
V. Skala, M. Smolik, and L. Karlicek. HS-Patch: A new Hermite smart bicubic patch modification. International Journal of Mathematics and Computers in Simulation 8:292-299, 2014. http://www.naun.org/main/NAUN/mcs/2014/a282002-086.pdf.
Wikipedia contributors. Kronecker product. Wikipedia, The Free Encyclopedia, 2021. https://en.wikipedia.org/wiki/Kronecker_product. [Accessed: 7 Oct 2021].
Wikipedia contributors. Multilinear polynomial. Wikipedia, The Free Encyclopedia, 2021. https://en.wikipedia.org/wiki/Multilinear_polynomial. [Accessed: 2 Oct 2021].
Wikipedia contributors. Tensor product. Wikipedia, The Free Encyclopedia, 2021. https://en.wikipedia.org/wiki/Tensor_product. [Accessed: 7 Oct 2021].
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