Constrained modeling of multi-sided patches

Abstract

We investigate genuinely multi-sided patches that interpolate ribbon surfaces along their boundaries. Recent works suggest defining patches over parametric domains with curved boundaries and hole loops, where the domain mimics the shape of the surface to be constructed. Cross-derivatives of the input are interpreted with respect to this curved parametric domain, but it is an open question how to initialize and modify these vector functions.We propose algorithms to set the cross-derivatives of multi-sided patches defined by Bézier and B-spline ribbons. Boundaries and surface constraints are inherited from adjacent patches, and our goal is to define a nice surface while ensuring smooth (G1) connections. We exploit that ribbon parameterizations induce ‘proportional’ cross-derivative magnitudes in 3D, and express cross-derivatives as the combination of vector functions and appropriately chosen scalar functions.Continuity constraints imply complex relations between the control points of the ribbons, so their direct modification is not feasible. Instead we suggest a constrained editing technique based on control vectors that significantly simplifies this task.