Free-form splines combining NURBS and basic shapes

We show how to combine, into one unified spline complex, C2 tensorproduct bi-cubic NURBS and G2 bi-cubic rational splines. The G2 splines are capable of exactly representing basic shapes such as (pieces of) quadrics and surfaces of revolution, including tori and cyclides. The main challenge is to bridge the differing continuity. We transform the G2 splines to splines that are C2 in homogeneous space. This yields Hermite data for a transitional strip of tensor-product splines of degree (6,5) that guarantees overall curvature continuity. Key to the construction is the C2 parameterization of circles in homogeneous space. We also cover the simpler G1 to C1 transition.