The simplest subdivision scheme for smoothing polyhedra (original) (raw)
C1 Smoothing of Polyhedra with Implicit Surface Patches
1991
Polyhedral "smoothing" is an efficient construction scheme for generating complex boundary models of solid physical objects. This paper presents efficient algorithms for generating families of curved solid objects with boundary topology relaled to the input polyhedron. Individual facets of a polyhedron are replaced by low degree implicit algebraic surface patches with local support. These quintic patches replace the CO contacts of planar facets with C l continuity along all interpatch boundaries. Selection of suitable instances of implicit surfaces as well as local control of the individual surface patches are achieved via simultaneous interpolation and weighted least-squares approximation. Asymptotic degree bounds are also given for the lowest degree implicitly defined, algebraic splines required La CI-smooth the vertices, edges, and facets of an arbitrary polyhedron in three dimensional real space ]R3. ·Supported in part by NSF gran~GGR 90-00028 and AFOSR contract 91-0276 2 C' CONTINUITY AND COMPATIBILITY CONDITIONS
A smooth subdivision scheme for hexahedral meshes
The Visual Computer, 2001
In a landmark paper, Catmull and Clark described a simple generalization of the subdivision rules for bi-cubic B-splines to arbitrary quadrilateral surface meshes. This smooth subdivision scheme has become a mainstay of surface modeling systems. Joy and MacCracken described a generalization of this surface scheme to volume meshes. Unfortunately, little is known about the smoothness and regularity of this scheme
Smooth Subdivision of Tetrahedral Meshes
2007
Abstract We describe a new subdivision scheme for unstructured tetrahedral meshes. Previous tetrahderal schemes based on generalizations of box splines have encoded arbitrary directional preferences in their associated subdivision rules that were not reflected in tetrahderal base mesh. Our method avoids this choice of preferred directions resulting a scheme that is simple to implement via repeated smoothing.
Ternary subdivision for quadrilateral meshes
Computer Aided Geometric Design, 2007
A well-documented problem of Catmull and Clark subdivision surfaces is that, in the neighborhood of extraordinary points, the curvature is unbounded and fluctuates. In fact, since one of the eigenvalues that determines elliptic shape is too small, the limit surface can have a saddle point when the designer's input mesh suggests a convex shape. Here, we replace, near the extraordinary point, Catmull-Clark subdivision by another set of rules based on refining each bi-cubic B-spline into nine. This provides many localized degrees of freedom for special rules that need not reach out to neighbor vertices in order to tune the behavior. In this paper, we provide a strategy for setting such degrees of freedom and exhibit tuned ternary quad subdivision that yields surfaces with bounded curvature, nonnegative weights and full contribution of elliptic and hyperbolic shape components.
Smoothing polyhedra using implicit algebraic splines
ACM SIGGRAPH Computer Graphics, 1992
Polyhedron "smoothing" is an efficient construction scheme for generating complex boundary models of solid physical objects. This paper presents efficient algorithms for generating families of curved solid objects with boundaty topology related to an input polyhedron. Individual faces of a polyhedron are replaced by low degree implicit algebraic surface patches with local support. These quintic patches replace the @ contacts of planar facets with C' continuity along all irtterpatch boundaries. Selection of suitable instances of implicit surfaces as well as local control of the individual surface patches are achieved via simultaneouss interpolation and weighted least-squares approximation.
A Unified Interpolatory Subdivision Scheme for Quadrilateral Meshes
ACM Transactions on Graphics, 2013
For approximating subdivision schemes, there are several unified frameworks for effectively constructing subdivision surfaces generalizing splines of an arbitrary degree. In this article, we present a similar unified framework for interpolatory subdivision schemes. We first decompose the 2n-point interpolatory curve subdivision scheme into repeated local operations. By extending the repeated local operations to quadrilateral meshes, an efficient algorithm can be further derived for interpolatory surface subdivision. Depending on the number n of repeated local operations, the continuity of the limit curve or surface can be of an arbitrary order CL, except in the surface case at a limited number of extraordinary vertices where C1 continuity with bounded curvature is obtained. Boundary rules built upon repeated local operations are also presented.
Smooth Low Degree Approximations of Polyhedra
1994
We present efficient algorithms to construct both C 1 and C 2 smooth meshes of cubic and quintic A-patches to approximate a given polyhedron P in three dimensions. The A~patch is a smooth and single-sheeted zero-contour patch of a trivariate polynomial in Bernstein-Bezier (BB) form defined within a tetrahedron. The smooth mesh constructions rely on a novel scheme to build an inner simplicial hull E consisting of tetrahedra and defined by the faces of the given polyhedron p, A single cubic or quintic A-patch is then constructed within each tetrahedron of the simplicial hull :E with the resulting surface being C l or C 2 smooth, respectively. The free parameters of each individual A-patch can be independently controlled to achieve both local and globa1shape deformations and a family of C 1 or C 2 smooth approximations of the original polyhedron.
Computer Graphics Forum, 2003
In this paper we introduce a new subdivision operator that unifies triangular and quadrilateral subdivision schemes. Designers often want the added flexibility of having both quads and triangles in their models. It is also well known that triangle meshes generate poor limit surfaces when using a quad scheme, while quad-only meshes behave poorly with triangular schemes. Our new scheme is a generalization of the well known Catmull-Clark and Loop subdivision algorithms. We show that our surfaces are C 1 everywhere and provide a proof that it is impossible to construct a C 2 scheme at the quad/triangle boundary. However, we provide rules that produce surfaces with bounded curvature at the regular quad/triangle boundary and provide optimal masks that minimize the curvature divergence elsewhere. We demonstrate the visual quality of our surfaces with several examples.
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 .
Non-uniform local interpolatory subdivision surfaces for regular quadrilateral meshes
2010
Subdivision is a powerful mechanism for generating curves and surfaces from discrete sets of control points. So far, the main advantage of subdivision methods with respect to other free-form representations, such as splines, has been acknowledged in their ability to generate smooth surfaces of arbitrary topology. In this paper we propose a method to generate non uniform subdivision surfaces interpolating regular quadrilateral meshes. We show that, choosing a suitable parameterization and properly setting edge and face point rules, these surfaces favorable compare both with their uniform counterpart and with non uniform tensor product splines.