In computer graphics, the terms parametric continuity (Ck) and geometric continuity (Gn) were introduced by Brian Barsky, to show that the smoothness of a curve can be measured either with respect to a particular parametrization or after allowing changes in the speed with which the parameter traces out the curve.

Parametric continuity

Parametric continuity (Ck) is a concept applied to parametric curves is said to be of class Ck are taken to be one-sided derivatives).

As a practical application of this concept, a curve describing the motion of an object with a parameter of time has C1 continuity when its velocity varies continuously, and C2 continuity when its acceleration varies continuously. For smoother motion, such as that of a camera's path while making a film, higher orders of parametric continuity may be required.

Order of parametric continuity

The various orders of parametric continuity can be described as follows:

• : zeroth derivative is continuous (curves are continuous)

• : zeroth and first derivatives are continuous

• : zeroth, first and second derivatives are continuous

• -th derivatives are continuous

Geometric continuity

A curve or surface being an increasing measure of smoothness. Consider the segments on either side of a point on a curve:

• : The curves touch at the join point.

• : The curves also share a common tangent direction at the join point.

• : The curves also share a common center of curvature at the join point.

parametric continuity. A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.

continuity are:

is constrained to be positive. (i.e., the direction, but not necessarily the magnitude, of the two vectors is equal).

continuity to appear smooth, for good aesthetics, such as those aspired to in architecture and sports car design, higher levels of geometric continuity are required. For example, class A surface or higher continuity to ensure smooth reflections in a car body.

A continuity. The same is true for a continuity is required, then cubic splines are typically chosen; these curves are frequently used in industrial design.

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