10,389 research outputs found
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 BĂ©zier 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 BĂ©zier 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 BĂ©zier 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
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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
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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
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Filling polygonal holes with bicubic patches
Consider a bicubic rectangular patch complex which surrounds an n-sided hole in R3. Then the problem of filling the hole with n bicubic rectangular patches is studied
Progressive surface modeling scheme from unorganised curves
This paper presents a novel surface modelling scheme to construct a freeform surface
progressively from unorganised curves representing the boundary and interior characteristic curves.
The approach can construct a base surface model from four ordinary or composite boundary curves
and support incremental surface updating from interior characteristic curves, some of which may not
be on the final surface. The base surface is first constructed as a regular Coons surface and upon receiving an interior curve sketch, it is then updated. With this progressive modelling scheme, a final
surface with multiple sub-surfaces can be obtained from a set of unorganised curves and transferred
to commercial surface modelling software for detailed modification. The approach has been tested
with examples based on 3D motion sketches; it is capable of dealing with unorganised design curves
for surface modelling in conceptual design. Its limitations have been discussed
An Intuitive Approach to Geometric Continuity for Parametric Curves and Surfaces (Extended Abstract)
The notion of geometric continuity is extended to an arbitrary order for curves and surfaces, and an intuitive development of constraints equations is presented that are necessary for it. The constraints result from a direct application of the univariate chain rule for curves, and the bivariate chain rule for surfaces. The constraints provide for the introduction of quantities known as shape parameters. The approach taken is important for several reasons: First, it generalizes geometric continuity to arbitrary order for both curves and surfaces. Second, it shows the fundamental connection between geometric continuity of curves and geometric continuity of surfaces. Third, due to the chain rule derivation, constraints of any order can be determined more easily than derivations based exclusively on geometric measures
Bivariate Hermite subdivision
A subdivision scheme for constructing smooth surfaces interpolating scattered data in 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 from which none of the pairs and with coincide, it is proved that the resulting surface (function) is . 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 if the data are not 'too irregular'. Finally the approximation properties of the methods are investigated
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