Abstract
In this article we study the problem of constructing an intermediate surface between two other surfaces defined by different iterative construction processes. This problem is formalised with Boundary Controlled Iterated Function System model. The formalism allows us to distinguish between subdivision of the topology and subdivision of the mesh. Although our method can be applied to surfaces with quadrangular topology subdivision, it can be used with any mesh subdivision (primal scheme, dual scheme or other.) Conditions that guarantee continuity of the intermediate surface determine the structure of subdivision matrices. Depending on the nature of the initial surfaces and coefficients of the subdivision matrices we can characterise the differential behaviour at the connection points between the initial surfaces and the intermediate one. Finally we study the differential behaviour of the constructed surface and show the necessary conditions to obtain an almost everywhere differentiable surface.
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References
Barnsley, M.: Fractals everywhere. Academic Press Professional, Inc., San Diego (1988)
Bensoudane, H.: Etude differentielle des formes fractales. PhD thesis, Université de Bourgogne (2009)
Bensoudane, H., Gentil, C., Neveu, M.: Fractional half-tangent of a curve described by iterated function system. Journal Of Applied Functional Analysis 4(2), 311–326 (2009)
Boumzaid, Y., Lanquetin, S., Neveu, M., Destelle, F.: An approximating-interpolatory subdivision scheme. Int. J. Pure Appl. Math. 71(1), 129–147 (2011)
Gentil, C.: Les fractales en synthèse d’image: le modèle IFS. PhD thesis, Université LYON 1 (1992)
Hutchinson, J.: Fractals and self-similarity. Indiana University Journal of Mathematics 30(5), 713–747 (1981)
Jiang, Q., Li, B., Zhu, W.: Interpolatory quad/triangle subdivision schemes for surface design. Computer Aided Geometric Design 26(8), 904–922 (2009)
Kosinka, J., Sabin, M., Dodgson, N.: Cubic subdivision schemes with double knots. Computer Aided Geometric Design (2012)
Levin, A., Levin, D.: Analysis of quasi uniform subdivision. Applied and Computational Harmonic Analysis 15(1), 18–32 (2003)
Daniel Mauldin, R., Williams, S.C.: Hausdorff dimension in graph directed constructions. Transactions of the American Mathematical Society 309(2), 811–829 (1988)
Podkorytov, S., Gentil, C., Sokolov, D., Lanquetin, S.: Geometry control of the junction between two fractal curves. Computer-Aided Design 45(2), 424–431 (2012); Solid and Physical Modeling (2012)
Prusinkiewicz, P., Hammel, M.: Language-Restricted Iterated Function Systems, Koch Constructions, and L-systems (1994)
Sokolov, D., Gentil, C., Bensoudane, H.: Differential behaviour of iteratively generated curves. In: Boissonnat, J.-D., Chenin, P., Cohen, A., Gout, C., Lyche, T., Mazure, M.-L., Schumaker, L. (eds.) Curves and Surfaces 2011. LNCS, vol. 6920, pp. 663–680. Springer, Heidelberg (2012)
Stam, J., Loop, C.: Quad/triangle subdivision. Computer Graphics Forum 22(1), 79–85 (2003)
Schaefer, S., Warren, J.: On c2 triangle/quad subdivision. ACM Trans. Graph. 24(1), 28–36 (2005)
Tosan, E., Bailly-Sallins, I., Gouaty, G., Stotz, I., Buser, P., Weinand, Y.: Une modélisation géométrique itérative basée sur les automates. In: GTMG 2006, Journées du Groupe de Travail en Modélisation Géométrique, Cachan, Mars 22-23, pp. 155–169 (2006)
Thollot, J., Tosan, E.: Constructive fractal geometry: constructive approach to fractal modeling using language operations (1995)
Zair, C.E., Tosan, E.: Fractal modeling using free form techniques. Comput. Graph. Forum 15(3), 269–278 (1996)
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Podkorytov, S., Gentil, C., Sokolov, D., Lanquetin, S. (2014). Joining Primal/Dual Subdivision Surfaces. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2012. Lecture Notes in Computer Science, vol 8177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54382-1_23
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DOI: https://doi.org/10.1007/978-3-642-54382-1_23
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