Dynamic NURBS swung surfaces for physics-based shape design
Given two plane curves, \({\mathbf c}_ 1(u)\) in the (x-z)plane and \({\mathbf c}_ 2(v)\) in the (x-y)plane and a continuous function \(\alpha(t)\), the authors define the swung surface as \({\mathbf s}= ({\mathbf c}_{1x}(u){\mathbf c}_{2x}(v), {\mathbf c}_{1x}(u) {\mathbf c}_{2y}(v), {\mathbf c}_{1z}(u))\). Then they assume that the curves are given as nonuniform rational B-splines (NURBS) depending on control points \({\mathbf p}(t)\) and weight \(w(t)\). For good measure they replace \({\mathbf c}_ 2\) by \(\alpha(t){\mathbf c}_ 2\). They develop a matrix formulation of all this (the matrix formulation in (3) is inconsistent but one understands easily what is meant; the basic formulas (4), (9) are immediate consequences of Euler's formula for homogeneous functions). The main idea is to use a model of thin shells under stress to apply finite element methods to compute the control data of the surface; using the matrix formulation this can then be done by conjugate gradients. Some practical points of implementation are discussed and examples given.
scientific article; zbMATH DE number 739603