Taxonomy of interpolation constraints on recursive subdivision surfaces (original) (raw)

Designing Catmull–Clark subdivision surfaces with curve interpolation constraints

Computers & Graphics, 2002

Generating subdivision surfaces with curve interpolation constraints is needed in both computer graphics and geometric modeling applications. In the context of the Doo-Sabin subdivision scheme, this can be achieved through the use of polygonal complexes as suggested by Nasri (Presented at the Fifth Siam Conference on Geometric Design, Nashville, 1997; Comput. Aided Geom. Des. 17 (2000) 595). A polygonal complex is simply a polygonal mesh whose structure depends on the subdivision scheme used and whose limit of subdivision is a curve rather than a surface. The subdivision scheme applied to these complexes is basically the same applied to the mesh defining the surface but with possible modification of its subdivision rules. The advantage of that lies in the retention of the same subdivision coefficients, thus saving the need for any further analysis at the limit. In this paper, we propose a method for using polygonal complexes to generate Catmull-Clark subdivision surfaces with curve interpolation constraints. The polygonal complexes are embedded here in the given mesh, which can possibly interpolate intersecting curves.

Smoothing and Near-Interpolatory Subdivision Surfaces

2004

In this paper we describe methods for computing smoothing and near-interpolatory (variational) subdivided surfaces, including those surfaces that meet data to within prescribed tolerances. The theory is based on standard results from constrained optimization combined with existing variational interpolatory subdivision schemes. The particular functional considered later in the paper is a discretization of the thin-plate spline functional. This paper considers the characterization and computational algorithms – numerical and smoothness issues will be deferred to future work. §

Subdivision for C¹ Surface Interpolation

2007

Interpolatory subdivision schemes generating surfaces from initial control nets with the topology of a regular grid are introduced and analyzed. The schemes are based upon the butterfly-scheme and are shown to have components with continuous first order derivatives. The generalization of these schemes for the application to general control nets is also discussed.

Progressive Interpolation based on Catmull-Clark Subdivision Surfaces

Computer Graphics Forum, 2008

We introduce a scheme for constructing a Catmull-Clark subdivision surface that interpolates the vertices of a quadrilateral mesh with arbitrary topology. The basic idea here is to progressively modify the vertices of an original mesh to generate a new control mesh whose limit surface interpolates all vertices in the original mesh. The scheme is applicable to meshes with any size and any topology, and it has the advantages of both a local scheme and a global scheme.

An alternative method for constructing interpolatory subdivision from approximating subdivision

Computer Aided Geometric Design, 2012

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy Computer Aided Geometric Design 29 (2012) 474-484 Contents lists available at SciVerse ScienceDirect

Interpolatory convexity-preserving subdivision schemes for curves and surfaces

Computer-Aided Design, 1992

Interpolatory convexity-preserving subdivision schemes for curves and surfaces are introduced, and a convergence analysis is presented. The schemes are defined by geometric constructions, and they are nonlinear in the control points. It is shown, by geometry-based proofs, that the limit curves and surfaces are C 1. interpolation, convexity, subdivision, curves, surfaces Only a few of the multitude of methods for surface interpolation treat the problem of convexity preservation 1, and then only for data on regular meshes. Recently, Dahlberg and Johansson 2 have studied the problem of convexity-preserving approximation by local schemes based on piecewise polynomials over patches. They prove the nonexistence of a local nontrivial convexity-preserving approximation operator, and they argue that this implies that a convexity-preserving procedure must take into account particular properties of the data. This paper suggests subdivision procedures for convex interpolation of curves and surfaces. The procedures are local, and they are applicable only to convex data sets. The analysis of the convergence of subdivision schemes and the analysis of the limit curves and surfaces have recently been developed considerably for schemes that are linear in the data points 3-7. However, the schemes presented in this paper are geometric and nonlinear. The

Tangent driven interpolative subdivision

Computers & Graphics, 2007

This paper explores a new family of 2D interpolating subdivision schemes that calculate subdivision points from tangent specifications. In this approach each level of subdivision drives the curve toward its specified tangents. The paper describes the underlying algorithmic formulation, and outlines a number of variations, which differ in how subdivision points are computed from tangents or in how tangents are derived. The different characteristics of these variations are shown to provide an interesting range of shapes, giving a high degree of artistic control. Each variation is shown to guarantee C1 continuity except for special identifiable cases, and some of these variations can produce circles. A number of examples are explored that demonstrate the power of the approach as a modeling tool for constructing complex curves, and a comparison is made with curves produced by the Kobbelt subdivision scheme.

An Approximating-Interpolatory Subdivision Scheme

Pure and Applied Mathematics

In the last decade, study and construction of quad/triangle subdivision schemes have attracted attention. The quad/triangle subdivision starts with a control mesh consisting of both quads and triangles and produces finer and finer meshes with quads and triangles (Figure 1). Designers often want to model certain regions with quad meshes and others with triangle meshes to get better visual quality of subdivision surfaces. Smoothness analysis tools exist for regular quad/triangle vertices. Moreover C 1 and C 2 quad/triangle schemes (for regular vertices) have been constructed. But to our knowledge, there are no quad/triangle schemes that unifies approximating and interpolatory subdivision schemes. In this paper we introduce a new subdivision operator that unifies triangular and quadrilateral subdivision schemes. Our new scheme is a generalization of the well known Catmull-Clark and Butterfly subdivision algorithms. We show that in the regular case along the quad/triangle boundary where vertices are shared by two adjacent quads and three adjacent triangles our scheme is C 2 everywhere except for ordinary Butterfly where our scheme is C 1 .

Interpolating Multiple Intersecting Curves Using Catmull-Clark Subdivision Surfaces

Computer-Aided Design and Applications, 2004

The problem of constructing a smooth subdivision surface interpolating multiple intersecting curves was partially addressed in the literature. In the context of Doo-Sabin subdivision surfaces, Nasri[3] presented a solution to interpolate unlimited number of curves through an extraordinary point. In the Catmull-Clark setting, no more than two intersecting curves could so far be interpolated. That is, the interpolation of multiple intersecting curves remains a non-trivial and elusive problem. This paper puts forth a solution to this problem. The solution relies in a fundamental way on the by-now well-known notion of Catmull-Clark Polygonal Complexes introduced in [5].

A 4-point interpolatory subdivision scheme for curve design

Computer Aided Geometric Design, 1987

A Cpoint interpolatory subdivision scheme with a tension parameter is analysed. It is shown that for a certain range of the tension parameter the resulting curve is C'. The role of the tension parameter is demonstrated by a few examples. The application to surfaces and some further potential generalizations are discussed.