G1 and G2 Surface interpolation over Curve Meshes and its Shape Control

A popular method to represent complicated surfaces is to interpolate the curve meshes that are defined by the boundary curves of the surface. The curve meshes we consider consist of Bezier curves, rational Bezier curves or composite curves depending on the shape we design. To interpolate these curve meshes smoothly, we use Gregory patches, rational boundary Gregory patches, general boundary Gregory patches and C2-Gregory patches as surface representations. One of the advantages of these surface representations is that cross boundary derivatives can be independently defined for each boundary. This enables us to create smooth surface shapes even on irregular curve meshes. In addition to smooth surface interpolation, these Gregory patches enable intuitive modification of their shapes with new control point layouts based on the cross boundary derivatives. Designers can easily modify surface shapes without losing G1/G2 continuitybetween adjacent patches by moving control points.