Rational Blending Functions Research Articles

Blending Bézier patch for multi-sided surface modeling

This study proposes a new n-sided (n≥4) control point-based surface patch, the blending Bézier patch (BB patch), by constructing corner Bézier surfaces and using Gregory corner blending. A BB patch is defined on a regular polygonal domain with an n-sided control net generalized from quadrilateral Bézier grids, and it reduces to the same degree as a Bézier patch when n=4. There are two main steps for constructing a BB patch: defining a corner Bézier patch for each corner and blending all corner Bézier patches using rational blending functions. Because the boundary behaviors of the BB patch are similar to those of the Bézier patch, a BB patch can be easily joined to the surrounding Bézier and other BB patches. As an application, we used BB patches to fill the holes with G2 continuity.

Construction of polynomial preserving cochain extensions by blending

A classical technique to construct polynomial preserving extensions of scalar functions defined on the boundary of an n n simplex to the interior is to use so-called rational blending functions. The purpose of this paper is to generalize the construction by blending to the de Rham complex. More precisely, we define polynomial preserving extensions which map traces of k k -forms defined on the boundary of the simplex to k k -forms defined in the interior. Furthermore, the extensions are cochain maps, i.e., they commute with the exterior derivative.

An iterative approach to barycentric rational Hermite interpolation

In this paper we study an iterative approach to the Hermite interpolation problem, which first constructs an interpolant of the function values at $$n+1$$ nodes and then successively adds m correction terms to fit the data up to the mth derivatives. In the case of polynomial interpolation, this simply reproduces the classical Hermite interpolant, but the approach is general enough to be used in other settings. In particular, we focus on the family of rational Floater–Hormann interpolants, which are based on blending local polynomial interpolants of degree d with rational blending functions. For this family, the proposed method results in rational Hermite interpolants, which depend linearly on the data, with numerator and denominator of degree at most $$(m+1)(n+1)-1$$ and $$(m+1)(n-d)$$ , respectively. They converge at the rate of $$O(h^{(m+1)(d+1)})$$ as the mesh size h converges to 0. After deriving the barycentric form of these interpolants, we prove the convergence rate for $$m=1$$ and $$m=2$$ , and show that the approximation results compare favourably with other constructions.

Multisided generalisations of Gregory patches

We propose two generalisations of Gregory patches to faces of any valency by using generalised barycentric coordinates in combination with two kinds of multisided Bézier patches. Our first construction builds on S-patches to generalise triangular Gregory patches. The local construction of Chiyokura and Kimura providing G1 continuity between adjoining Bézier patches is generalised so that the novel Gregory S-patches of any valency can be smoothly joined to one another. Our second construction makes a minor adjustment to the generalised Bézier patch structure to allow for cross-boundary derivatives to be defined independently per side. We show that the corresponding blending functions have the inherent ability to blend ribbon data much like the rational blending functions of Gregory patches. Both constructions take as input a polygonal mesh with vertex normals and provide G1 surfaces interpolating the input vertices and normals. Due to the full locality of the methods, they are well suited for geometric modelling as well as computer graphics applications relying on hardware tessellation.

Ribbon-based transfinite surfaces

One major issue in CAGD is to model complex objects using free-form surfaces of general topology. A natural approach is curvenet-based design, where designers directly create and modify feature curves. These are interpolated by smoothly connected, multi-sided patches, which can be represented by transfinite surfaces, defined as a combination of side interpolants or ribbons. A ribbon embeds Hermite data, i.e., prescribed positional and cross-derivative functions along boundary curves.The paper focuses on two transfinite schemes: the first is an enhanced and extended variant of a multi-sided generalization of the classical Coons patch (Várady et al., 2011); the second one is based on a new concept of combining doubly curved composite ribbons, each one interpolating three adjacent sides. Main contributions include various ribbon parameterizations that surpass former methods in quality and computational efficiency. It is proven that these surfaces smoothly interpolate the prescribed ribbon data. Both formulations are based on non-regular convex polygonal domains and distance-based, rational blending functions. A few examples illustrate the results.

A negative experiment with univariate blending functions

Univariate rational blending functions are investigated in this paper. It is shown that the usual Kronecker delta condition is not sufficient to define visually pleasing interpolants. The set of functions taken in this paper satisfy this condition; however, they produce small identations in the interpolant where it looks as if a thumb had pushed the curve. An insight into the geometric structure of the interpolation explains this negative effect.

Curves under tension

We introduce a new type of parametrically defined space curve via rational blending functions involving parameters which allow for convenient control over local shape attributes while maintaining global curvature continuity.