Multisided generalisations of Gregory patches

Abstract

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 rst construction builds on S-patches to generalise triangular Gregory patches. The local construction of Chiyokura and Kimura providing G 1 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 dened 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 G 1 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.

Colour interpolants for polygonal gradient meshes

“…Generalised Bézier patches [27,15,22], or GB patches, are multisided control point patches that generalise tensor-product Bézier patches to a patch with an arbitrary number of sides. Although these patches can be of any degree, in this paper we are only interested in the cubic variant.…”

“…This then completes the GB patch definition. The Gregory generalised Bézier patch [15] or generalised Gregory patch is a simple extension of the ordinary GB patch and generalises the quadrilateral Gregory patch [7, Section 5.1]. The boundary curves and ribbons are given as cubic Bézier curves and surfaces, respectively.…”

“…We call these types of meshes generalised gradient meshes. We compare the existing topologically unrestricted gradient mesh [20] and the cubic mean value interpolant [19], as well as the Gregory generalised Bézier patch [15] and the Charrot-Gregory corner interpolator patch [6], both of which we make suitable for fully local, per-patch, globally G 1 smooth colour interpolation.…”

Splines for Meshes with Irregularities

“…Note that the Venn diagram regions for smooth constructions overlap since constructions can borrow features from each other: for example, [29] shows that "barycentric" and the "Gregory" approach are not mutually exclusive, and Section 4.3 will suggest that it makes sense to combine a few steps of (accelerated, guided) subdivision with a G-spline construction.…”

Optimization for Parameters of Generalized Bézier Patch

“…GB 面片虽然相对于 S 面片等控制点的数量有 极大的改进, 但是在数值计算方面还有些许不足, 尤其是局部参数(将非负重心坐标参数化 [12][13] )的 计算. 无论是改进后的 GB 面片 [2,14] 还是在应用 [15] 中, 都没有充分简化参数. 本文对 GB 面片的局部 参数进行优化, 并证明优化的局部参数生成的 GB 面片边界具有与文献 [1]相同的拼接条件.…”