zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A unified framework for primal/dual quadrilateral subdivision schemes. (English) Zbl 0969.68155
Summary: Quadrilateral subdivision schemes come in primal and dual varieties, splitting faces or respectively vertices. The scheme of Catmull-Clark is an example of the former, while the Doo-Sabin scheme exemplifies the latter. In this paper we consider the construction of an increasing sequence of alternating primal/dual quadrilateral subdivision schemes based on a simple averaging approach. Beginning with a vertex split step we successively construct variants of Doo-Sabin and Catmull-Clark schemes followed by novel schemes generalizing B-splines of bidegree up to nine. We prove the schemes to be $C^{1}$ at irregular surface points, and analyze the behavior of the schemes as the number of averaging steps increases. We discuss a number of implementation issues common to all quadrilateral schemes. In particular we show how both primal and dual quadrilateral schemes can be implemented in the same code, opening up new possibilities for more flexible geometric modeling applications and $p$-versions of the Subdivision Element Method. Additionally we describe a simple algorithm for adaptive subdivision of dual schemes.

MSC:
68U05Computer graphics; computational geometry
WorldCat.org
Full Text: DOI
References:
[1] Biermann, H.; Levin, A.; Zorin, D.: Piecewise smooth subdivision surfaces with normal control. Proceedings of SIGGRAPH 2000, 113-120 (2000)
[2] Catmull, E.; Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-aided design 10, No. 6, 350-355 (1978)
[3] Cavaretta, A.S., Dahmen, W., Micchelli, C.A. Stationary subdivision. Mem. Amer. Math. Soc. 93 (453) · Zbl 0741.41009
[4] Cirak, F.; Ortiz, M.; Schröder, P.: Subdivision surfaces: A new paradigm for thin-shell finite-element analysis. Int. J. Numer. meth. Engng. 47, 2039-2072 (2000) · Zbl 0983.74063
[5] Cohen, E.; Lyche, T.; Riesenfeld, R.: Discrete b-splines and subdivision techniques in computer aided geometric design and computer graphics. Computer graphics and image processing 14, No. 2, 87-111 (1980)
[6] Doo, D.; Sabin, M.: Analysis of the behaviour of recursive division surfaces near extraordinary points. Computer-aided design 10, No. 6, 356-360 (1978)
[7] Dyn, N.; Levin, D.; Gregory, J. A.: A butterfly subdivision scheme for surface interpolation with tension control. ACM trans. Graph. 9, No. 2, 160-169 (1990) · Zbl 0726.68076
[8] Habib, A.; Warren, J.: Edge and vertex insertion for a class of c1 subdivision surfaces. Computer aided geometric design 16, No. 4, 223-247 (1999) · Zbl 0916.68151
[9] Jury, E. I.: Theory and applications of the z-transform method. (1964)
[10] Kobbelt, L.: Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Proceedings of eurographics 96, computer graphics forum, 409-420 (1996)
[11] Kobbelt, L.: 3 subdivision. Proceedings of SIGGRAPH 2000, 103-112 (2000)
[12] Loop, C.: Smooth subdivision surfaces based on triangles, master’s thesis. (1987)
[13] Peters, J.; Reif, U.: The simplest subdivision scheme for smoothing polyhedra. ACM trans. Graph. 16, No. 4, 420-431 (1997)
[14] Prautzsch, H.: Smoothness of subdivision surfaces at extraordinary points. Adv. comput. Math. 9, No. 3--4, 377-389 (1998) · Zbl 0918.65094
[15] Qu, R.: Recursive subdivision algorithms for curve and surface design, ph.d. Thesis. (1990)
[16] Reif, U.: A unified approach to subdivision algorithms near extraordinary points. Computer aided geometric design 12, 153-174 (1995) · Zbl 0872.65007
[17] Samet, H.: The design and analysis of spatial data structures. (1990) · Zbl 0719.90022
[18] Stam, J.: On subdivision schemes generalizing uniform b-spline surfaces of arbitrary degree. Computer aided geometric design 18, No. 5, 383-396 (2001) · Zbl 0970.68184
[19] Velho, L.: Using semi-regular 4--8 meshes for subdivision surfaces. Journal of graphics tools (2001) · Zbl 0971.68176
[20] Velho, L.; Zorin, D.: 4--8 subdivision. Computer aided geometric design 18, No. 5, 397-427 (2001) · Zbl 0969.68157
[21] Warren, J., Weimer, H., 2001. Subdivision for geometric design, to appear · Zbl 0970.68177
[22] Xu, X.; Kondo, K.: Adaptive refinements in subdivision surfaces. Eurographics 99 Proceedings (1999)
[23] Zorin, D.: A method for analysis of c1-continuity of subdivision surfaces. SIAM J. Numer. anal. 37, No. 4, 1677-1708 (2000) · Zbl 0959.65021
[24] Zorin, D.: Smoothness of subdivision on irregular meshes. Constructive approximation 16, No. 3, 359-397 (2000) · Zbl 0965.65024
[25] Zorin, D.; Schröder, P.; Sweldens, W.: Interpolating subdivision for meshes with arbitrary topology. Proceedings of SIGGRAPH 96, 189-192 (1996)