Interpolatory Schemes

All the previous examples have the property that the corners are smoothed off the original polygon. It is also possible for a scheme to have the property that the new polygon has new v-vertices which lie exactly at the corresponding old polygon vertices. If this is true after one step it will also be true after two, or three, or more, and indeed it is true, by induction, of the limit curve. 

The scheme illustrated here has its v-vertices at the original vertices, and each of its e-vertices at the place defined by a parametric cubic Lagrange interpolant through four points. The weights for an e-vertex are -1/16, 9/16, 9/16 and -1/16, and the scheme is therefore known as the four-point scheme. 

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N.Dyn, D.Levin and J.Gregory A 4-point interpolatory scheme for curve design. CAGD 4(4) pp257-268, 1987... 

M. Marinov, N.Dyn amd D.Levin Geometrically controlled 4-point interpolatory scheme. pp301-315 in Advances in Multiresolution for Geometric Modelhng (eds Dodgson, Floater, Sabin), Springer 2005 ISBN 3-540-21462-3... 

Equation (4.127) is written in a form that is suitable for the prediction of the properties of mixtures from independent evaluations, experimental or otherwise, of the thermal conductivity of the pure components, kq. It is, in fact, written in such a way that it automatically reproduces the pure component values and acts, therefore, as an interpolatory scheme between them which has its basis in a well-characterized approximation of the rigorous kinetic theory. However, further approximations are almost always required before the Hirschfelder-Eucken equation can be implemented. 

N. Dyn, D.Levin and A.Luzzato Non-stationary interpolatory subdivision schemes reproducing spaces of exponential polynomials. Found Comput Math, ppl97-206, 2003... 

L.Romani A circle-preserving C2 Hermite interpolatory subdivision scheme with tension control. CAGD 27(1), pp36-47, 2010 C.Deng and G.Wang Incenter subdivision scheme for curve interpolation. CAGD 27(1), pp48-59, 2010... 

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