Constructive modeling of G1 bifurcation

This paper deals with the modeling of G1 bifurcation. A branch-blending strategy is applied to model the bifurcation such that the coherence among the individual branching segments is characterized. To achieve this, bi-cubic Bezier patches are first used to generate three half-tubular surfaces by sweeping operations. The bifurcation modeling is then converted to a problem of filling two triangular holes surrounded by the swept half-tubular surfaces. In order for the bifurcation model to be G1, the candidate surfaces for hole filling are required to have (i) an inter-patch tangential continuity along the so-called star-lines; and (ii) a cross-boundary tangential continuity with the surrounding half-tubular surfaces. For inter-patch tangential continuity, we use the method proposed by Gregory and Zhou (1994) to determine the center-point, the star-lines, and the associated vector-valued cross-boundary derivatives. The problem of ensuring the cross-boundary tangential continuity with the surrounding surfaces is more difficult. We first derive the conditions for the twist-compatibility from the requirements of cross-boundary tangential continuity. The solutions for the conditions derived are then developed. The hole boundaries, originally in cubic Bezier form, are constructively modified to the quintic form to ensure the twist-compatibility and uniqueness of the tangent planes at the hole corners. Subsequently, the half-tubular surfaces in bi-cubic Bezier form are degree-elevated to quintic form along the sweeping directions. This is followed by the modification of the second row control points with the half-tubular surfaces in order to retain a cubic form of the surfaces' cross-boundary derivatives. Vector-valued cross-boundary derivatives in quintic Bezier form are constructed for the hole patches. Using a Coons-Boolean sum approach, the derivatives are utilized to modify the three Bezier patches into the final bi-quintic form in the triangle fill area for each hole.