Interpolating Multiple Intersecting Curves Using Catmull-Clark Subdivision Surfaces (original) (raw)
Related papers
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.
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.
Constructing an Interpolatory Subdivision Scheme from Doo-Sabin Subdivision
2011
This paper presents an interpolatory subdivision scheme derived from the Doo-Sabin subdivision scheme. We first present the relations among three curve subdivision schemes, namely a four point interpolatory subdivision scheme, a cubic B-spline curve subdivision scheme, and the Chaikin's algorithm that generates uniform quadratic B-spline curves. By generalizing these relations to the surface case, we derive an interpolatory surface subdivision scheme from the Doo-Sabin subdivision scheme, a generalization of the Chaikin's algorithm to surface subdivision. In the new subdivision scheme, we also introduce a variable tension parameter that is dependent to local control vertices. The variable tension parameter can be used to effectively control the resulting limit surface of the proposed subdivision scheme.
Open Journal of Applied Sciences, 2013
This paper presents a general formula for (2m + 2)-point n-ary interpolating subdivision scheme for curves for any integer m ≥ 0 and n ≥ 2 by using Newton interpolating polynomial. As a consequence, the proposed work is extended for surface case, which is equivalent to the tensor product of above proposed curve case. These formulas merge several notorious curve/surface schemes. Furthermore, visual performance of the subdivision schemes is also presented.
A simple method for interpolating meshes of arbitrary topology by Catmull-Clark surfaces
The Visual Computer, 2010
Interpolating an arbitrary topology mesh by a smooth surface plays important role in geometric modeling and computer graphics. In this paper we present an efficient new algorithm for constructing Catmull–Clark surface that interpolates a given mesh. The control mesh of the interpolating surface is obtained by one Catmull–Clark subdivision of the given mesh with modified geometric rule. Two methods—push-back operation based method and normal-based method—are presented for the new geometric rule. The interpolation method has the following features: (1) Efficiency: we obtain a generalized cubic B-spline surface to interpolate any given mesh in a robust and simple manner. (2) Simplicity: we use only simple geometric rule to construct control mesh for the interpolating subdivision surface. (3) Locality: the perturbation of a given vertex only influences the surface shape near this vertex. (4) Freedom: for each edge and face, there is one degree of freedom to adjust the shape of the limit surface. These features make interpolation using Catmull–Clark surfaces very simple and thus make the method itself suitable for interactive free-form shape design.
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. §
ACM Transactions on Graphics, 1990
In industrial design, the tool of choice for constructing surfaces that interpolate curves is the Boolean sum surface technique. However, if curves do not lie on constant parameter lines, reparametrizations will be needed, and this may introduce derivative discontinuities. A new technique which shows promise in overcoming this problem is described here. The method is based on describing the interpolation problem directly as a system of linear equations rather than as a curve-blending problem. The resulting system of equations is usually underdetermined and can be solved using numerical linear algebra methods without the a priori determination of certain parameters. The “free” parameters can be used to control the shape of the resulting surface. Two examples of the procedure are given.
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.
Taxonomy of interpolation constraints on recursive subdivision surfaces
The Visual Computer, 2002
This paper is the first of two, which together describe and classify the various situations that any complete study of interpolation constraints for a recursive subdivision surface needs to consider. They do so in the form of a systematic taxonomy of situations. Presented here are curve cases, which provide good illustrations of principles which will be used in both contexts; surfaces will be addressed in the second paper. Known results are classified and open questions identified.
Polygonal mesh regularization for subdivision surfaces interpolating meshes of curves
The Visual Computer, 2003
A recursive-subdivision surface interpolating a mesh of curves can be generated from a given polygonal mesh (or a polyhedron) and some tagged control polygons by constructing a polygonal complex for each of these polygons. This process will modify the geometry, and possibly the topology, of the polygonal mesh and could result in poorly shaped surfaces across the interpolated curves. The problem can be minimized by applying some fairing procedure that regularly repositions the vertices of the mesh. This paper provides an approach to regularize a polygonal mesh based on the Laplacian and mean curvature of the vertices. The results are useful, especially if further constraints such as normal or cross curvature are imposed across the interpolated curves, where more irregularity can be introduced on the polygonal mesh.