### Geometric Continuity of Parametric Curves

**Authors:**

Barsky, Brian A.

DeRose, Tony D.
**Technical Report Identifier:** CSD-84-205
**October 1984**

CSD-84-205.pdf

**Abstract:** 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 *C^n* or *n*th order parametric continuity. We show that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves.

We define *n*th order geometric continuity (*G^n*), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed form of parametric continuity. *G^n* 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.