Reconstruction of Surfaces and Surfaces- on- Surfaces from Unorganized Weighted Points

We present a unified approach for the reconstruction of both a C 1 smooth domain surface 3d and a G smooth function SI (surface-on-surface) on the domain surface 3d from an unorganized collection of weighted points {(Xi,y" Zi, Wi)}. The set of points P = {(Xi,Yi,Zi)} are assumed to be sampled from on or near an unknown domain S in m? while the weights Wi are assumed sampled from some unknown scalar function F on the domain S. Examples include the pressure F on the wing S of an airplane or the temperature T on a portion S of the human body. The simplicity and unified nature of our algorithm arises from several uses of appropriate sub-structures of the three-dimensional Dclaunay Triangulation, and its dual the three-dimensional Voronoi diagram. The C1 smooth surface Sd and the Cl smooth function SJ are constructed using trivariate cubic Bezier patches. We also present techniques for efficiently producing different visualizations of the reconstructed surface Sd and the surface-on-surface SJ. "This work was supported in part by NSF grants CCR 92-22467, DMS 91-01424, AFOSR grants F49620-93-10138, F49620.94-1.0080, NASA grant NAG-1-H73 and a gift from AT&T.

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