Browse Reports

Parametric spline curves are typically constructed so that the first n parametric derivatives agree where the curve segments abut. This type of continuity condition has become known as Cn or nth order geometric continuity. It has previously bee n shown that the use of parametric continuity disallows many parameterizations which generate geometrically smooth curves. a relaxed form of nth order parametric continuity has been developed and dubbed nth order geometric continuity and denoted Gn. These notes explore three characterizations of geometric continuity. First, the concept of equivalent parameterizations is used t o view geometric continuity as a measure of continuity that is parameterization independent, that is, a measure that is invariant under reparameterization. The second characterization develops necessary and sufficient conditions, called Beta-constraints, for geometric continuity of curves. Finally, the third characterization shows that two curves meet with Gn continuity if and only if their arc length parameterizations meet with Cn continuity. Gn continuity provides for the introduction of n quantities known as shape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices. Several applications of geometric continuity are present. First, composite Bezier curves are stitched together with G1 and G2 continuity using geometric constructions. Then, a subclass of the Catmull-Rom splines based on geometric continuity and possessing shape parameters is discussed. Finally, quadratic G1 and cubic G2 Beta-splines are developed using the geometric constructions for the geometrically continuous Bezier segments.