Construction of smooth branching surfaces using T-splines

Abstract

The request for designing or reconstructing objects from planar cross sections arises in various applications, ranging from CAD to GIS and Medical Imaging. The present work focuses on the “one-to-many” branching problem, where one of the planes can be populated with many, possibly tortuous and densely packed, contours. The proposed method combines the proximity information offered by the Euclidean Voronoi diagram with the concept of surrounding curve, introduced in Gabrielides et al. (2007), and T-splines technology Sederberg et al. (2003) for securing a flexible and portable representation. Our algorithm delivers a single cubic T-spline that deviates from the given contours less than a user-specified tolerance, measured via the so-called discrete Fréchet distance Eiter and Mannila (1994) and is C2 everywhere except from a finite set of point-neighborhoods. Subject to minor enrichment, the algorithm is also capable to handle the “many-to-many” configuration as well as the global reconstruction problem involving contours on several planes.