We use cookies to ensure that we give you the best experience on our website. You can change your cookie settings at any time. Otherwise, we'll assume you're OK to continue.

Durham University

Research & business

View Profile

Publication details for Dr Ioannis Ivrissimtzis

Ivrissimtzis, Ioannis & Seidel, Hans-Peter (2004). Evolutions of Polygons in the Study of Subdivision Surfaces. Computing 72(1-2): 93-103.

Author(s) from Durham


We employ the theory of evolving n-gons in the study of subdivision surfaces. We show that for subdivision schemes with small stencils the eige¬nanalysis of an evolving polygon, corresponding either to a face or to the 1-¬ring neighborhood of a vertex, complements in a geometrically intuitive way the eigenanalysis of the subdivision matrix. In the applications, we study the types of singularities that may appear on a subdivision surface, and we find properties of the subdivision surface that depend on the initial control polyhedron only.



Bachmann, F., Schmidt, E.: n-Ecke. B.I.-Hochschultaschenbu¨ cher 1970.


Ball, A.A., Storry, D.J.T.: Conditions for tangent plane continuity over recursively generated
B-spline surfaces. ACM Transactions on Graphics 7, 83–102 (1988).


Berlekamp, E.R., Gilbert, E.N., Sinden, F.W.: A polygon problem. Am. Math. Mon. 72, 233–241


Bruckstein, A.M., Sapiro, G., Shaked, D.: Evolutions of planar polygons. IJPRAI 9, 991–1014


Darboux, M.G.: Sur un proble` me de ge` ome` trie e` le` mentaire. Bull. Sci. Math. 2, 298–304 (1878).


Davis, P.J.: Circulant matrices. Wiley-Interscience 1979.


Doo, D., Sabin, M.: Behaviour of recursive division surfaces near extraordinary points.
Computer Aided Design 10, 356–360 (1978).


Dyn, N.: Subdivision schemes in computer-aided geometric design. In: Light, W. (ed.): Advances
in numerical analysis, vol. 2, p. 36–104. Clarendon Press 1992.


Fisher, J.C., Jamison, R.E.: Properties of affinely regular polygons. Geom. Dedicata 69, 241–259


Kobbelt, L.:
p3 subdivision. In: Siggraph 00, Conference Proceedings, pp. 103–112, 2000.


Loop, C.T.: Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah,
Department of Mathematics 1987.


Peters, J., Reif, U.: The simplest subdivision scheme for smoothing polyhedra. ACM
Transactions on Graphics 16, 420–431 (1997).


Peters, J., Reif, U.: Analysis of algorithms generalizing B-spline subdivision. SIAM Journal on
Numerical Analysis 35, 728–748 (1998).


Prautzsch, H., Reif, U.: Necessary conditions for subdivision surfaces. Adv. Comput. Math. 10,
209–217 (1999).


Reif, U.: A unified approach to subdivision algorithms near extraordinary vertices. Computer
Aided Geometric Design 12, 153–174 (1995)


Zorin, D.: Ck Continuity of Subdivision Surfaces. PhD thesis, California Institute of Technology,
Department of Computer Science 1996.