## Non-stationary subdivision schemes for surface interpolation based on exponential polynomials.(English)Zbl 1195.65015

A subdivision scheme for generating curves and surfaces from a finite set of control points is proposed. The main fact is that the subdivision scheme is non-stationary: the mask used to compute the new points changes from level to level. The definition of the mask at each level goes as follows: Given some finite set of exponential polynomials (functions of the type $$x^\alpha e^{\beta x}$$) the mask is the one fitting a kind of butterfly-shaped stencil for the set of exponential polynomials. Thus, the computation of the mask at each level is equivalent to solve a linear system. Examples of how the algorithm works for parametric surfaces as torus and spheres are shown.
A careful analysis of the convergence and of the smoothness of the subdivision scheme is done proving that these non-stationary schemes have the same smoothness and approximation order as the classical butterfly interpolatory scheme.

### MSC:

 65D17 Computer-aided design (modeling of curves and surfaces) 65D10 Numerical smoothing, curve fitting
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### References:

 [1] Beccari, C.; Casciola, G.; Romani, L., A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics, Comput. aided geom. design, 24, 1-9, (2007) · Zbl 1171.65325 [2] Beccari, C.; Casciola, G.; Romani, L., An interpolating 4-point $$C^2$$ ternary non-stationary subdivision scheme with tension control, Comput. aided geom. design, 24, 210-219, (2007) · Zbl 1171.65326 [3] Cavaretta, A.; Dahmen, W.; Micchelli, C.A., Stationary subdivision, Mem. amer. math. soc., 93, 1-186, (1991) · Zbl 0741.41009 [4] Chaikin, G., An algorithm for high speed curve generation, Computer graphics and image processing, 3, 346-349, (1974) [5] Choi, Y.-J.; Lee, Y.-J.; Yoon, J.; Lee, B.-G.; Kim, Y.-J., A new class of non-stationary interpolatory subdivision schemes based on exponential polynomials, (), 563-570, (SpringerLink, PDF-file) · Zbl 1160.68616 [6] Cohen, E.; Lyche, T.; Riesenfeld, R., Discrete B-spline and subdivision techniques in computer-aided geometric design and computer graphics, Computer graphics and image processing, 14, 87-111, (1980) [7] Doo, D.; Sabin, M., Behaviour of recursively division surfaces near extraordinary points, Comput.-aided design, 10, 356-360, (1978) [8] Deslauriers, G.; Dubuc, S., Symmetric iterative interpolation, Constr. approx., 5, 49-68, (1989) · Zbl 0659.65004 [9] Dyn, N., Subdivision schemes in computer-aided geometric design, (), 36-104 · Zbl 0760.65012 [10] Dyn, N.; Gregory, J.A.; Levin, D., A four-point interpolatory subdivision scheme for curve design, Comput. aided geom. design, 4, 257-268, (1987) · Zbl 0638.65009 [11] Dyn, N.; Gregory, J.A.; Levin, D., A butterfly subdivision scheme for surface interpolation with tension control, ACM trans. graph., 9, 160-169, (1990) · Zbl 0726.68076 [12] Dyn, N.; Levin, D., Analysis of asymptotically equivalent binary subdivision schemes, J. of math. anal. appl., 193, 594-621, (1995) · Zbl 0836.65012 [13] Dyn, N.; Levin, D.; Luzzatto, A., Exponential reproducing subdivision scheme, Found. comp. math., 3, 187-206, (2003) · Zbl 1095.41001 [14] Dyn, N.; Levin, D.; Yoon, J., Analysis of univariate non-stationary subdivision schemes with application to Gaussian-based interpolatory schemes, SIAM J. math. anal., 39, 470-488, (2007) · Zbl 1132.41302 [15] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), John Hopkins University Press Baltimore · Zbl 0865.65009 [16] Jena, M.J.; Shunmugaraj, P.; Das, P.J., A subdivision algorithm for trigonometric spline curves, Comput. aided geom. design, 19, 71-88, (2002) · Zbl 0984.68165 [17] Jena, M.J.; Shunmugaraj, P.; Das, P.J., A non-stationary subdivision scheme for generalizing trigonometric surfaces to arbitrary meshes, Comput. aided geom. design, 20, 61-77, (2003) · Zbl 1069.65557 [18] Jena, M.K.; Shunmugaraj, P.; Das, P.C., A non-stationary subdivision scheme for curve interpolation, Anziam J., 44, E, 216-235, (2003) · Zbl 1078.65528 [19] Morin, G.; Warren, J.; Weimer, H., A subdivision scheme for surfaces of revolution, Comput. aided geom. design, 18, 483-502, (2001) · Zbl 0970.68177 [20] Romani, L., From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms, J. of comp. and appl. math., 224, 383-396, (2009) · Zbl 1159.65019 [21] Warren, J.; Weimer, H., Subdivision methods for geometric design, (2002), Academic Press New York
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