• An extended r -adaptive IGA framework is proposed for weak-discontinuous problems. • A Gaussian monitor redistributes geometric resolution using level-set interfaces. • Control-point relocation enhances interface resolution without mesh refinement. • Up to 65.7% error reduction is achieved with negligible computational overhead.

Currently, NURBS is the most popular mathematical tool for curve and surface design. One of the geometric meanings of NURBS curve is that the limiting of the curve is its regular control curve, when all weights tend to infinity. In this paper, we apply toric degeneration theory and knot insertion algorithm of NURBS curve to study the number of regular control curves of NURBS curve, and the upper bound of the number is presented. Moreover, some representative examples are illustrated to verify our results.

Abstract Toric surface patch is the multi‐sided generalization of classical Bézier surface patch. Geometric continuity of the parametric surface patches plays a crucial role in geometric modeling. In this paper, the necessary and sufficient conditions of curvature continuity between toric surface patches are illustrated with the theory of toric degeneration. Furthermore, some practical sufficient conditions of curvature continuity of toric surface patches are also developed.

Rational Bezier functions are widely used as mapping functions in surface reparameterization, finite element analysis, image warping and morphing. The injectivity (one-to-one property) of a mapping function is typically necessary for these applications. Toric Bezier patches are generalizations of classical patches (triangular, tensor product) which are defined on the convex hull of a set of integer lattice points. We give a geometric condition on the control points that we show is equivalent to the injectivity of every 2D toric Bezier patch with those control points for all possible choices of weights. This condition refines that of Craciun, et al., which only implied injectivity on the interior of a patch.

The control polygon of a rational Bézier curve is well-defined and has geometric significance; there is a sequence of weights under which the limiting position of the curve is the control polygon. For a rational Bézier surface patch, there are many possible polyhedral control structures, and none is canonical. We propose a not necessarily polyhedral control structure for rational surface patches, regular control surfaces, which are certain C 0 spline surfaces. While not unique, regular control surfaces are exactly the possible limiting positions of a rational Bézier patch when the weights vary.