Point-Normal Subdivision Curves and Surfaces

This paper proposes to generalize linear subdivision schemes to nonlinear subdivision schemes for curve and surface modeling by refining vertex positions together with refinement of unit control normals at the vertices. For each round of subdivision, new control normals are obtained by projections of linearly subdivided normals onto unit circle or sphere while new vertex positions are obtained by updating linearly subdivided vertices along the directions of the newly subdivided normals. Particularly, the new position of each linearly subdivided vertex is computed by weighted averages of end points of circular or helical arcs that interpolate the positions and normals at the old vertices at one ends and the newly subdivided normal at the other ends. The main features of the proposed subdivision schemes are three folds: (1) The point-normal (PN) subdivision schemes can reproduce circles, circular cylinders and spheres using control points and control normals; (2) PN subdivision schemes generalized from convergent linear subdivision schemes converge and can have the same smoothness orders as the linear schemes; (3) PN $C^2$ subdivision schemes generalizing linear subdivision schemes that generate $C^2$ subdivision surfaces with flat extraordinary points can generate visually $C^2$ subdivision surfaces with non-flat extraordinary points. Experimental examples have been given to show the effectiveness of the proposed techniques for curve and surface modeling.Comment: 30 pages, 17 figures, 22.5M

Isogeometric analysis using manifold-based smooth basis functions

We present an isogeometric analysis technique that builds on manifold-based smooth basis functions for geometric modelling and analysis. Manifold-based surface construction techniques are well known in geometric modelling and a number of variants exist. Common to all is the concept of constructing a smooth surface by blending together overlapping patches (or, charts), as in differential geometry description of manifolds. Each patch on the surface has a corresponding planar patch with a smooth one-to-one mapping onto the surface. In our implementation, manifold techniques are combined with conformal parametrisations and the partition-of-unity method for deriving smooth basis functions on unstructured quadrilateral meshes. Each vertex and its adjacent elements on the surface control mesh have a corresponding planar patch of elements. The star-shaped planar patch with congruent wedge-shaped elements is smoothly parameterised with copies of a conformally mapped unit square. The conformal maps can be easily inverted in order to compute the transition functions between the different planar patches that have an overlap on the surface. On the collection of star-shaped planar patches the partition of unity method is used for approximation. The smooth partition of unity, or blending functions, are assembled from tensor-product b-spline segments defined on a unit square. On each patch a polynomial with a prescribed degree is used as a local approximant. To obtain a mesh-based approximation scheme, the coefficients of the local approximants are expressed in dependence of vertex coefficients. This yields a basis function for each vertex of the mesh which is smooth and non-zero over a vertex and its adjacent elements. Our numerical simulations indicate the optimal convergence of the resulting approximation scheme for Poisson problems and near optimal convergence for thin-plate and thin-shell problems

Smooth Subdivision Surfaces: Mesh Blending and Local Interpolation

Subdivision surfaces are widely used in computer graphics and animation. Catmull-Clark subdivision (CCS) is one of the most popular subdivision schemes. It is capable of modeling and representing complex shape of arbitrary topology. Polar surface, working on a triangle-quad mixed mesh structure, is proposed to solve the inherent ripple problem of Catmull-Clark subdivision surface (CCSS). CCSS is known to be C1 continuous at extraordinary points. In this work, we present a G2 scheme at CCS extraordinary points. The work is done by revising CCS subdivision step with Extraordinary-Points-Avoidance model together with mesh blending technique which selects guiding control points from a set of regular sub-meshes (named dominative control meshes) iteratively at each subdivision level. A similar mesh blending technique is applied to Polar extraordinary faces of Polar surface as well. Both CCS and Polar subdivision schemes are approximating. Traditionally, one can obtain a CCS limit surface to interpolate given data mesh by iteratively solving a global linear system. In this work, we present a universal interpolating scheme for all quad subdivision surfaces, called Bezier Crust. Bezier Crust is a specially selected bi-quintic Bezier surface patch. With Bezier Crust, one can obtain a high quality interpolating surface on CCSS by parametrically adding CCSS and Bezier Crust. We also show that with a triangle/quad conversion process one can apply Bezier Crust on Polar surfaces as well. We further show that Bezier Crust can be used to generate hollowed 3D objects for applications in rapid prototyping. An alternative interpolating approach specifically designed for CCSS is developed. This new scheme, called One-Step Bi-cubic Interpolation, uses bicubic patches only. With lower degree polynomial, this scheme is appropriate for interpolating large-scale data sets. In sum, this work presents our research on improving surface smoothness at extraordinary points of both CCS and Polar surfaces and present two local interpolating approaches on approximating subdivision schemes. All examples included in this work show that the results of our research works on subdivision surfaces are of high quality and appropriate for high precision engineering and graphics usage

GPGPU-beschleunigte Reduktion bathymetrischer Geländemodelle mit T-Spline-Flächen

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New strategies for curve and arbitrary-topology surface constructions for design

This dissertation presents some novel constructions for curves and surfaces with arbitrary topology in the context of geometric modeling. In particular, it deals mainly with three intimately connected topics that are of interest in both theoretical and applied research: subdivision surfaces, non-uniform local interpolation (in both univariate and bivariate cases), and spaces of generalized splines. Specifically, we describe a strategy for the integration of subdivision surfaces in computer-aided design systems and provide examples to show the effectiveness of its implementation. Moreover, we present a construction of locally supported, non-uniform, piecewise polynomial univariate interpolants of minimum degree with respect to other prescribed design parameters (such as support width, order of continuity and order of approximation). Still in the setting of non-uniform local interpolation, but in the case of surfaces, we devise a novel parameterization strategy that, together with a suitable patching technique, allows us to define composite surfaces that interpolate given arbitrary-topology meshes or curve networks and satisfy both requirements of regularity and aesthetic shape quality usually needed in the CAD modeling framework. Finally, in the context of generalized splines, we propose an approach for the construction of the optimal normalized totally positive (B-spline) basis, acknowledged as the best basis of representation for design purposes, as well as a numerical procedure for checking the existence of such a basis in a given generalized spline space. All the constructions presented here have been devised keeping in mind also the importance of application and implementation, and of the related requirements that numerical procedures must satisfy, in particular in the CAD context