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Register a new userCharacterisation of rational and NURBS developable surfaces in Computer Aided Design
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Added by
Leonardo Fernandez-Jambrina
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Authors
Leonardo Fernandez-Jambrina
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In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions $Lambda$, $M$, $ u$. 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 $Lambda$, $M$, $ u$, 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 $Lambda$, $M$, $ u$ . The results are readily extended to rational spline developable surfaces.
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In this paper we construct developable surface patches which are bounded by two rational or NURBS curves, though the resulting patch is not a rational or NURBS surface in general. This is accomplished by reparameterizing one of the boundary curves. The reparameterization function is the solution of an algebraic equation. For the relevant case of cubic or cubic spline curves, this equation is quartic at most, quadratic if the curves are Bezier or splines and lie on parallel planes, and hence it may be solved either by standard analytical or numerical methods.
In this paper we address the issue of designing developable surfaces with Bezier patches. We show that developable surfaces with a polynomial edge of regression are the set of developable surfaces which can be constructed with Aumanns algorithm. We also obtain the set of polynomial developable surfaces which can be constructed using general polynomial curves. The conclusions can be extended to spline surfaces as well.
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