Characterisation of rational and NURBS developable surfaces in Computer Aided Design

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.

💡 Research Summary

This paper presents a comprehensive characterization of rational developable surfaces—surfaces with zero Gaussian curvature—within the framework of Computer Aided Design (CAD) that relies on NURBS (Non‑Uniform Rational B‑Splines). The authors start from the geometric definition of a developable surface as the envelope of a one‑parameter family of planes. For each parameter λ the surface satisfies both a(λ)·x + b(λ)=0 and its derivative a′(λ)·x + b′(λ)=0, guaranteeing that every ruling (straight line) lies in a common tangent plane.

The central contribution is the reformulation of the classic developability condition (d(t) − c(t))·(c′(t)×d′(t)) = 0 in terms of blossom (polar form) representations of the bounding curves c(t) and d(t). By expressing the curves in homogeneous 4‑D coordinates (including weights) and using the de Casteljau algorithm, the authors identify four points pₙ₋₁⁰(t), pₙ₋₁¹(t), qₙ₋₁⁰(t), qₙ₋₁¹(t) that must be coplanar. Coplanarity is equivalent to a linear dependence relation that can be written with three rational functions Λ(t), M(t), and σ(t):

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