An algorithm to improve parameterizations of rational Bézier surfaces using rational bilinear reparameterization

Optimizing conformality of NURBS surfaces by general bilinear transformations

“…In practice, the algorithm produces significantly better surface parameterizations than the methods presented in [36,38]. Furthermore, the degree of the resultant surface is the same as the surfaces generated by [38]. The main contributions of this paper are as follows.…”

“…The coefficients of the initial general bilinear transformation is obtained by approximating the conformal mapping of its 3D discretized mesh using a least square method, which is further optimized numerically by the Levenberg-Marquardt method [40]. In practice, the algorithm produces significantly better surface parameterizations than the methods presented in [36,38]. Furthermore, the degree of the resultant surface is the same as the surfaces generated by [38].…”

“…The Möbius transformations as well as the rational bilinear transformations presented in [36,38] always map the rectangle parameter domain to itself. As a result, the four boundary curves as well as the orthogonality at the four corners remain unchanged during the transformations.…”

“…Furthermore, the rational bilinear reparameterization coefficients are determined by a trivial interpolation method, which is only suitable for a special surface case. To obtain more uniform and orthogonal iso-parametric curves for rational Bézier surfaces, Yang et al [38] presented an optimization algorithm to minimize the nonlinear energy measuring uniformity and orthogonality deviations using the rational bilinear reparameterizations, which produces a better parameterization with the cost of degree elevation. However both the Möbius transformation and rational bilinear transformation presented in [36,38] suffer from the fact that they always map the four surface boundary curves to themselves in the transformation.…”

Fast global and partial reflective symmetry analyses using boundary surfaces of mechanical components

“…Parameterization or reparameterization of edges can be addressed with dedicated algorithms [27,28,29,30,31] but closed curves add the problem of defining appropriately their origin (see Figure 3),…”