Subdivision surfaces are considered which consist of trior quadrilateral patchesin a mostly regul... more Subdivision surfaces are considered which consist of trior quadrilateral patchesin a mostly regular arrangement with finitely many irregularities. A sharp estimateon the lowest possible degree of the patches is given. It depends on thesmoothness and flexibility of the underlying subdivision scheme.KeywordsSubdivision, piecewise polynomial surface, arbitrary topology, extraordinarypoint.1 IntroductionThis paper deals with subdivision algorithms acting on arbitrary 2D control nets as
Piecewise polynomial subdivision surfaces are considered which consist of tri-or quadrilateral pa... more Piecewise polynomial subdivision surfaces are considered which consist of tri-or quadrilateral patches in a mostly regular arrangement with finitely many irregularities. A sharp estimate on the lowest possible degree of the polynomial patches is given. It depends on the smoothness and flexibility of the underlying subdivision scheme.
We construct a uniformly stable family of bases for tensor product spline approximation on domain... more We construct a uniformly stable family of bases for tensor product spline approximation on domains in R d. These bases are derived from the standard B-spline basis by normalization with respect to the L p-norm and a selection process relying on refined estimates for the de Boor-Fix functionals.
We examine the smoothness properties of the principal curvatures of subdivision surfaces near irr... more We examine the smoothness properties of the principal curvatures of subdivision surfaces near irregular points. In particular we give an estimate of their L p class based on the eigenstructure of the subdivision matrix. As a result we can show that the popular Loop and Catmull-Clark schemes (among many others) have square integrable principal curvatures enabling their use as shape functions in FEM treatments of the thin shell equations.
We derive Sobolev-type inner products with respect to which hat functions on arbitrary triangulat... more We derive Sobolev-type inner products with respect to which hat functions on arbitrary triangulations of domains in R d are orthogonal. Compared with linear interpolation, the resulting approximation schemes yield superior accuracy at little extra cost.
A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algor... more A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algorithms is presented. The main challenge here is the verification of injectivity of the characteristic map. The tools are sufficiently versatile and easy to wield to allow, as an application, a full analysis of algorithms generalizing biquadratic and bicuble B-spline subdivision. In the case of generalized biquadratic subdivision the analysis yields a hitherto unknown sharp hound strictly less than one on the second largest eigenvalue of any smoothly converging subdivision.
We describe and analyze a multigrid algorithm for finite element approximations of second order e... more We describe and analyze a multigrid algorithm for finite element approximations of second order elliptic boundary value problems with weighted extended b-splines (web-splines). This new technique provides high accuracy with relatively low-dimensional subspaces, does not require any grid generation, and is ideally suited for hierarchical solution techniques. In particular, we show that the standard W-cycle yields uniform convergence, i.e., the required number of iterations is bounded independent of the grid width.
We show that the interpolatory four-point-scheme with tension parameter ω generates C 1-limit cur... more We show that the interpolatory four-point-scheme with tension parameter ω generates C 1-limit curves if and only if 0 < ω < ω * , where ω * ≈ 0.19273 is the unique real solution of the cubic equation 32ω 3 + 4ω − 1 = 0.
Journal of Mathematical Analysis and Applications, 1999
Let s be the graph of a function s: ⍀ ª ޒ over the bounded set ⍀ ; ޒ 2. The fairness of s can... more Let s be the graph of a function s: ⍀ ª ޒ over the bounded set ⍀ ; ޒ 2. The fairness of s can be measured by means of functionals depending on the principal curvatures , and the area a of s. Here we consider functionals of type 1 2 1, ϱ , p 0 functions can also be restricted by additional constraints involving in particular point and gradient data.
We specify conditions on degenerate triangular B~zier patches which guarantee curvature continuit... more We specify conditions on degenerate triangular B~zier patches which guarantee curvature continuity, i.e., the existence of a regular C2-reparametrization. Further, it is shown how N/> 3 patches satisfying these conditions can be joined around a common point.
We establish an exhaustive catalogue of 42 equivalence classes of bivariate quadratic surfaces in... more We establish an exhaustive catalogue of 42 equivalence classes of bivariate quadratic surfaces in affine n-space. Here two surfaces belong to the same class if they differ only by zero components or by affine transformations of domain and range.
Given a polynomial p of degree n we want to find a best L 2-approximation over the unit interval ... more Given a polynomial p of degree n we want to find a best L 2-approximation over the unit interval from polynomials of degree m < n. This problem is shown to be equivalent to the problem of finding the best Euclidean approximation of the vector of Bernstein-Bézier coefficients of p from the vector of degree-raised Bernstein-Bézier coefficients of polynomials of degree m.
The construction of weighted extended B-splines (web-splines), as recently introduced by the auth... more The construction of weighted extended B-splines (web-splines), as recently introduced by the authors and J. Wipper for uniform knot sequences, is generalized to the nonuniform case. We show that web-splines form a stable basis for splines on arbitrary domains in R m which provides optimal approximation power. Moreover, homogeneous boundary conditions, as encountered frequently in finite element applications, can be satisfied exactly by using an appropriate weight function. To illustrate the performance of the method, it is applied to a scattered data fitting problem and a finite element approximation of an elliptic boundary value problem.
For subdivision surfaces, it is important to characterize local shape near flat spots and points ... more For subdivision surfaces, it is important to characterize local shape near flat spots and points where the surface is not twice continuously differentiable. Applying general principles derived in [PR0x], this paper characterizes shape near such points for the subdivision schemes devised by Catmull and Clark and by Loop. For generic input data, both schemes fail to converge to the hyperbolic or elliptic limit shape suggested by the geometry of the input mesh: the limit shape is a function of the valence of the extraordinary point rather than the mesh geometry. We characterize the meshes for which the schemes behave as expected and indicate modifications of the schemes that prevent convergence to the wrong shape. We also introduce a type of chart that, for a specific scheme, can help a designer to detect early when a mesh will lead to undesirable curvature behavior.
Special issue on class A surfaces Class A surfaces are those outside parts of industrial objects ... more Special issue on class A surfaces Class A surfaces are those outside parts of industrial objects which attribute to their aesthetic appeal. The most important class A surfaces are those representing hoods, roofs, etc. of automobiles. The design of class A surfaces thus involves not only functional criteria but also subjective ones relating to style and appearance. Creating tools which support the work of a stylist is a challenging task in the areas of CAD and CAGD. Design criteria and tools involve Curvature distribution of feature curves, Simulation of reflection lines, Reflection line based modeling, Surface fitting with aesthetic constraints, Reverse engineering to meet styling intent. CAGD invites papers addressing these problems and related ones. All articles will be reviewed according to the standard practice of CAGD. All articles must be submitted to Elseviers e-submission site to the attention of G. Farin and indicate clearly that the submission is intended for the Class A surfaces Special Issue.
Given a polyhedron, construct a new polyhedron by connecting every edgemidpoint to its four neigh... more Given a polyhedron, construct a new polyhedron by connecting every edgemidpoint to its four neighboring edge-midpoints. This refinement rule yields a a 1 surface and the surface has a piecewise quadratic parametrization except at finite number of isolated points. We analyze and improve the construction.
Subdivision surfaces are considered which consist of trior quadrilateral patchesin a mostly regul... more Subdivision surfaces are considered which consist of trior quadrilateral patchesin a mostly regular arrangement with finitely many irregularities. A sharp estimateon the lowest possible degree of the patches is given. It depends on thesmoothness and flexibility of the underlying subdivision scheme.KeywordsSubdivision, piecewise polynomial surface, arbitrary topology, extraordinarypoint.1 IntroductionThis paper deals with subdivision algorithms acting on arbitrary 2D control nets as
Piecewise polynomial subdivision surfaces are considered which consist of tri-or quadrilateral pa... more Piecewise polynomial subdivision surfaces are considered which consist of tri-or quadrilateral patches in a mostly regular arrangement with finitely many irregularities. A sharp estimate on the lowest possible degree of the polynomial patches is given. It depends on the smoothness and flexibility of the underlying subdivision scheme.
We construct a uniformly stable family of bases for tensor product spline approximation on domain... more We construct a uniformly stable family of bases for tensor product spline approximation on domains in R d. These bases are derived from the standard B-spline basis by normalization with respect to the L p-norm and a selection process relying on refined estimates for the de Boor-Fix functionals.
We examine the smoothness properties of the principal curvatures of subdivision surfaces near irr... more We examine the smoothness properties of the principal curvatures of subdivision surfaces near irregular points. In particular we give an estimate of their L p class based on the eigenstructure of the subdivision matrix. As a result we can show that the popular Loop and Catmull-Clark schemes (among many others) have square integrable principal curvatures enabling their use as shape functions in FEM treatments of the thin shell equations.
We derive Sobolev-type inner products with respect to which hat functions on arbitrary triangulat... more We derive Sobolev-type inner products with respect to which hat functions on arbitrary triangulations of domains in R d are orthogonal. Compared with linear interpolation, the resulting approximation schemes yield superior accuracy at little extra cost.
A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algor... more A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algorithms is presented. The main challenge here is the verification of injectivity of the characteristic map. The tools are sufficiently versatile and easy to wield to allow, as an application, a full analysis of algorithms generalizing biquadratic and bicuble B-spline subdivision. In the case of generalized biquadratic subdivision the analysis yields a hitherto unknown sharp hound strictly less than one on the second largest eigenvalue of any smoothly converging subdivision.
We describe and analyze a multigrid algorithm for finite element approximations of second order e... more We describe and analyze a multigrid algorithm for finite element approximations of second order elliptic boundary value problems with weighted extended b-splines (web-splines). This new technique provides high accuracy with relatively low-dimensional subspaces, does not require any grid generation, and is ideally suited for hierarchical solution techniques. In particular, we show that the standard W-cycle yields uniform convergence, i.e., the required number of iterations is bounded independent of the grid width.
We show that the interpolatory four-point-scheme with tension parameter ω generates C 1-limit cur... more We show that the interpolatory four-point-scheme with tension parameter ω generates C 1-limit curves if and only if 0 < ω < ω * , where ω * ≈ 0.19273 is the unique real solution of the cubic equation 32ω 3 + 4ω − 1 = 0.
Journal of Mathematical Analysis and Applications, 1999
Let s be the graph of a function s: ⍀ ª ޒ over the bounded set ⍀ ; ޒ 2. The fairness of s can... more Let s be the graph of a function s: ⍀ ª ޒ over the bounded set ⍀ ; ޒ 2. The fairness of s can be measured by means of functionals depending on the principal curvatures , and the area a of s. Here we consider functionals of type 1 2 1, ϱ , p 0 functions can also be restricted by additional constraints involving in particular point and gradient data.
We specify conditions on degenerate triangular B~zier patches which guarantee curvature continuit... more We specify conditions on degenerate triangular B~zier patches which guarantee curvature continuity, i.e., the existence of a regular C2-reparametrization. Further, it is shown how N/> 3 patches satisfying these conditions can be joined around a common point.
We establish an exhaustive catalogue of 42 equivalence classes of bivariate quadratic surfaces in... more We establish an exhaustive catalogue of 42 equivalence classes of bivariate quadratic surfaces in affine n-space. Here two surfaces belong to the same class if they differ only by zero components or by affine transformations of domain and range.
Given a polynomial p of degree n we want to find a best L 2-approximation over the unit interval ... more Given a polynomial p of degree n we want to find a best L 2-approximation over the unit interval from polynomials of degree m < n. This problem is shown to be equivalent to the problem of finding the best Euclidean approximation of the vector of Bernstein-Bézier coefficients of p from the vector of degree-raised Bernstein-Bézier coefficients of polynomials of degree m.
The construction of weighted extended B-splines (web-splines), as recently introduced by the auth... more The construction of weighted extended B-splines (web-splines), as recently introduced by the authors and J. Wipper for uniform knot sequences, is generalized to the nonuniform case. We show that web-splines form a stable basis for splines on arbitrary domains in R m which provides optimal approximation power. Moreover, homogeneous boundary conditions, as encountered frequently in finite element applications, can be satisfied exactly by using an appropriate weight function. To illustrate the performance of the method, it is applied to a scattered data fitting problem and a finite element approximation of an elliptic boundary value problem.
For subdivision surfaces, it is important to characterize local shape near flat spots and points ... more For subdivision surfaces, it is important to characterize local shape near flat spots and points where the surface is not twice continuously differentiable. Applying general principles derived in [PR0x], this paper characterizes shape near such points for the subdivision schemes devised by Catmull and Clark and by Loop. For generic input data, both schemes fail to converge to the hyperbolic or elliptic limit shape suggested by the geometry of the input mesh: the limit shape is a function of the valence of the extraordinary point rather than the mesh geometry. We characterize the meshes for which the schemes behave as expected and indicate modifications of the schemes that prevent convergence to the wrong shape. We also introduce a type of chart that, for a specific scheme, can help a designer to detect early when a mesh will lead to undesirable curvature behavior.
Special issue on class A surfaces Class A surfaces are those outside parts of industrial objects ... more Special issue on class A surfaces Class A surfaces are those outside parts of industrial objects which attribute to their aesthetic appeal. The most important class A surfaces are those representing hoods, roofs, etc. of automobiles. The design of class A surfaces thus involves not only functional criteria but also subjective ones relating to style and appearance. Creating tools which support the work of a stylist is a challenging task in the areas of CAD and CAGD. Design criteria and tools involve Curvature distribution of feature curves, Simulation of reflection lines, Reflection line based modeling, Surface fitting with aesthetic constraints, Reverse engineering to meet styling intent. CAGD invites papers addressing these problems and related ones. All articles will be reviewed according to the standard practice of CAGD. All articles must be submitted to Elseviers e-submission site to the attention of G. Farin and indicate clearly that the submission is intended for the Class A surfaces Special Issue.
Given a polyhedron, construct a new polyhedron by connecting every edgemidpoint to its four neigh... more Given a polyhedron, construct a new polyhedron by connecting every edgemidpoint to its four neighboring edge-midpoints. This refinement rule yields a a 1 surface and the surface has a piecewise quadratic parametrization except at finite number of isolated points. We analyze and improve the construction.
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Papers by Ulrich Reif