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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
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
Intersection of three-dimensional geometric surfaces
Calculating the line of intersection between two three-dimensional objects and using the information to generate a third object is a key element in a geometry development system. Techniques are presented for the generation of three-dimensional objects, the calculation of a line of intersection between two objects, and the construction of a resultant third object. The objects are closed surfaces consisting of adjacent bicubic parametric patches using Bezier basis functions. The intersection determination involves subdividing the patches that make up the objects until they are approximately planar and then calculating the intersection between planes. The resulting straight-line segments are connected to form the curve of intersection. The polygons in the neighborhood of the intersection are reconstructed and put back into the Bezier representation. A third object can be generated using various combinations of the original two. Several examples are presented. Special cases and problems were encountered, and the method for handling them is discussed. The special cases and problems included intersection of patch edges, gaps between adjacent patches because of unequal subdivision, holes, or islands within patches, and computer round-off error
Filling n-sided regions with G1 triangular Coons B-spline patches
International audienceFilling n-sided regions is an essential operation in shape and surface modeling. Positional and tangential continuities are highly required in designing and manufacturing. We propose a method for filling n-sided regions with untrimmed triangular Coons B-spline patches, preserving G1 continuity exactly. The algorithm first computes a central point, a central normal, the central, and the corner derivative vectors. Then the region is split into n triangular areas by connecting the central point to each corner of the boundary. These inner curves and all cross-boundary derivatives are computed fulfilling G1 compatibility conditions. And finally, the triangular patches are generated in the Coons B-spline form, one boundary of which is regressed to the central vertex. Neither positional nor tangential error is introduced by this method. And only one degree elevation is needed
Connectivity Control for Quad-Dominant Meshes
abstract: Quad-dominant (QD) meshes, i.e., three-dimensional, 2-manifold polygonal meshes comprising mostly four-sided faces (i.e., quads), are a popular choice for many applications such as polygonal shape modeling, computer animation, base meshes for spline and subdivision surface, simulation, and architectural design. This thesis investigates the topic of connectivity control, i.e., exploring different choices of mesh connectivity to represent the same 3D shape or surface. One key concept of QD mesh connectivity is the distinction between regular and irregular elements: a vertex with valence 4 is regular; otherwise, it is irregular. In a similar sense, a face with four sides is regular; otherwise, it is irregular. For QD meshes, the placement of irregular elements is especially important since it largely determines the achievable geometric quality of the final mesh.
Traditionally, the research on QD meshes focuses on the automatic generation of pure quadrilateral or QD meshes from a given surface. Explicit control of the placement of irregular elements can only be achieved indirectly. To fill this gap, in this thesis, we make the following contributions. First, we formulate the theoretical background about the fundamental combinatorial properties of irregular elements in QD meshes. Second, we develop algorithms for the explicit control of irregular elements and the exhaustive enumeration of QD mesh connectivities. Finally, we demonstrate the importance of connectivity control for QD meshes in a wide range of applications.Dissertation/ThesisDoctoral Dissertation Computer Science 201
A survey of partial differential equations in geometric design
YesComputer aided geometric design is an area
where the improvement of surface generation techniques
is an everlasting demand since faster and more accurate
geometric models are required. Traditional methods
for generating surfaces were initially mainly based
upon interpolation algorithms. Recently, partial differential
equations (PDE) were introduced as a valuable
tool for geometric modelling since they offer a number
of features from which these areas can benefit. This work
summarises the uses given to PDE surfaces as a surface
generation technique togethe
Multisided B-spline Patches Over Extraordinary Regions
We propose a generalised B-spline construction that extends uniform bi-degree B-splines to multisided regions spanned over extraordinary regions in quad-dominant meshes. We show how the structure of the existing cubic multisided B-spline patch can be generalised to work with B-spline basis functions of arbitrary degree and can be spanned over extraordinary vertices as well as extraordinary faces of quad-dominant meshes. The resulting multisided surfaces are Cd-1 continuous internally and connect with Gd-1 continuity to adjacent regular and other multisided B-splines patches. In addition, we design several specialised functions that increase the visual quality of the patches, in both the extraordinary vertex and face settings.<br/
A multisided C-2 B-spline patch over extraordinary vertices in quadrilateral meshes
We propose a generalised B-spline construction that extends uniform bicubic B-splines to multisided regions spanned over extraordinary vertices in quadrilateral meshes. We show how the structure of the generalised Bezier patch introduced by Varady et al. can be adjusted to work with B-spline basis functions. We create ribbon surfaces based on B-splines using special basis functions. The resulting multisided surfaces are C-2 continuous internally and connect with G(2) continuity to adjacent regular and other multisided B-splines patches. We visually assess the quality of these surfaces and compare them to Catmull-Clark limit surfaces on several challenging geometrical configurations. (C) 2020 The Author(s). Published by Elsevier Ltd
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