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Volume 39, Issue 4
Characterisation of Rational and NURBS Developable Surfaces in Computer Aided Design

Leonardo Fernández-Jambrina

DOI: 10.4208/jcm.2003-m2019-0226

J. Comp. Math., 39 (2021), pp. 556-573.

Published online: 2021-05

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  • Abstract

In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions $Λ, M, ν.$ Properties of developable surfaces are revised in this framework. In particular, a closed algebraic formula for the edge of regression of the surface is obtained in terms of the functions $Λ, M, ν,$ which are closely related to the ones that appear in the standard decomposition of the derivative of the parametrisation of one of the bounding curves in terms of the director vector of the rulings and its derivative. It is also shown that all rational developable surfaces can be described as the set of developable surfaces which can be constructed with a constant $Λ, M, ν .$ The results are readily extended to rational spline developable surfaces.

  • Keywords

NURBS, Bézier, Rational, Spline, NURBS, Developable surfaces.

  • AMS Subject Headings

65D17, 68U07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-39-556, author = {Fernández-Jambrina , Leonardo}, title = {Characterisation of Rational and NURBS Developable Surfaces in Computer Aided Design}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {39}, number = {4}, pages = {556--573}, abstract = {

In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions $Λ, M, ν.$ Properties of developable surfaces are revised in this framework. In particular, a closed algebraic formula for the edge of regression of the surface is obtained in terms of the functions $Λ, M, ν,$ which are closely related to the ones that appear in the standard decomposition of the derivative of the parametrisation of one of the bounding curves in terms of the director vector of the rulings and its derivative. It is also shown that all rational developable surfaces can be described as the set of developable surfaces which can be constructed with a constant $Λ, M, ν .$ The results are readily extended to rational spline developable surfaces.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2003-m2019-0226}, url = {http://global-sci.org/intro/article_detail/jcm/19150.html} }
TY - JOUR T1 - Characterisation of Rational and NURBS Developable Surfaces in Computer Aided Design AU - Fernández-Jambrina , Leonardo JO - Journal of Computational Mathematics VL - 4 SP - 556 EP - 573 PY - 2021 DA - 2021/05 SN - 39 DO - http://doi.org/10.4208/jcm.2003-m2019-0226 UR - https://global-sci.org/intro/article_detail/jcm/19150.html KW - NURBS, Bézier, Rational, Spline, NURBS, Developable surfaces. AB -

In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions $Λ, M, ν.$ Properties of developable surfaces are revised in this framework. In particular, a closed algebraic formula for the edge of regression of the surface is obtained in terms of the functions $Λ, M, ν,$ which are closely related to the ones that appear in the standard decomposition of the derivative of the parametrisation of one of the bounding curves in terms of the director vector of the rulings and its derivative. It is also shown that all rational developable surfaces can be described as the set of developable surfaces which can be constructed with a constant $Λ, M, ν .$ The results are readily extended to rational spline developable surfaces.

Leonardo Fernández-Jambrina. (2021). Characterisation of Rational and NURBS Developable Surfaces in Computer Aided Design. Journal of Computational Mathematics. 39 (4). 556-573. doi:10.4208/jcm.2003-m2019-0226
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