G2 Surface Interpolation Over General Topology Curve Networks

The basic idea of curve network-based design is to construct smoothly connected surface patches, that interpolate boundaries and cross-derivatives extracted from the curve network. While the majority of applications demands only tangent plane (G(1)) continuity between the adjacent patches, curvature continuous connections (G(2)) may also be required. Examples include special curve network configurations with supplemented internal edges, master-slave curvature constraints, and general topology surface approximations over meshes. The first step is to assign optimal surface curvatures to the nodes of the curve network; we discuss different optimization procedures for various types of nodes. Then interpolant surfaces called parabolic ribbons are created along the patch boundaries, which carry first and second derivative constraints. Our construction guarantees that the neighboring ribbons, and thus the respective transfinite patches, will be G(2) continuous. We extend Gregory's multi-sided surface scheme in order to handle parabolic ribbons, involving the blending functions, and a new sweepline parameterization. A few simple examples conclude the paper.