Polygonal patches of high order continuity

A polygonal patch is defined to fill an n-sided hole within a rectangular Ck patch framework. First a reparameterization of the surface around the hole is constructed, that is defined outside a regular polygon. The polygonal patch is an interpolant, defined inside the polygon, that matches this parameterization up to order k along the boundary. Some modifications and handles to control the shape of the patch are described

Geometric continuous patch complexes

A theory of geometric continuity of arbitrary order is presented. Conditions of geometric continuity at a vertex where a number of patches meet are investigated. Geometric continuous patch complexes are introduced as the appropriate setting for the representation of surfaces in CAGD. The theory is applied to the modelling of closed surfaces and the fitting of triangular patches into a geometric continuous patch complex

A pentagonal surface patch for computer-aided geometric design

A vector valued interpolation scheme for a pentagon is described which is compatible with surface patches which have a rectangular domain of definition. Such a scheme could be useful in computer- aided geometric design problems, where a pentagonal patch occurs within a rectangular patch framework

High order continuous polygonal patches

A polygonal patch method is described which can be used to fill a polygonal hole within a given k'th order continuous rectangular patch complex. The method is relatively easy to implement, since it only re- quires Ck extensions of the rectangular patch complex defined in terms of the rectangular patch parameterizations. The method is illustrated by reference to C2 bicubic B-spline surfaces

Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization

In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries consisting of a set of B-spline curves. Instead of forming the computational domain by a simple boundary, planar domains with high genus and more complex boundary curves are considered. Firstly, some pre-processing operations including B\'ezier extraction and subdivision are performed on each boundary curve in order to generate a high-quality planar parameterization; then a robust planar domain partition framework is proposed to construct high-quality patch-meshing results with few singularities from the discrete boundary formed by connecting the end points of the resulting boundary segments. After the topology information generation of quadrilateral decomposition, the optimal placement of interior B\'ezier curves corresponding to the interior edges of the quadrangulation is constructed by a global optimization method to achieve a patch-partition with high quality. Finally, after the imposition of C1=G1-continuity constraints on the interface of neighboring B\'ezier patches with respect to each quad in the quadrangulation, the high-quality B\'ezier patch parameterization is obtained by a C1-constrained local optimization method to achieve uniform and orthogonal iso-parametric structures while keeping the continuity conditions between patches. The efficiency and robustness of the proposed method are demonstrated by several examples which are compared to results obtained by the skeleton-based parameterization approach

Smooth parametric surfaces and n-sided patches

The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed

Bivariate Hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in $\mathbb{R}^3$ is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points $\{(x_i, y_i)\}_{i=1}^N$ from which none of the pairs $(x_i,y_i)$ and $(x_j,y_j)$ with $i\neq j$ coincide, it is proved that the resulting surface (function) is $C^1$. The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is $C^2$ if the data are not 'too irregular'. Finally the approximation properties of the methods are investigated