Geometric Continuity of Parametric Curves

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 parametric continuity. We show that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves. We define nth order geometric continuity (Gn), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed for of parametric 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 the theory are discussed, along with topics of future research.