Parametric continuity

Parametric continuity is a concept applied to parametric curves describing the smoothness of the parameter's value with distance along the curve.

Definition

A curve can be said to have "C"ⁿ continuity if

:$frac\{d^\{n\}s\}\{dt^\{n$

is continuous of value throughout the curve.

As an example of a practical application of this concept, a curve describing the motion of an object with a parameter of time, must have "C"¹ continuity for the object to have finite acceleration. For smoother motion, such as that of a camera's path while making a film, higher levels of parametric continuity are required.

Order of continuity

[
Bézier curve segments attached that is only C⁰ continuous.]

The various order of parametric continuity can be described as follows [ [http://www.cs.helsinki.fi/group/goa/mallinnus/curves/curves.html Parametric Curves ] ] :
* C⁰: curves are joined
* C¹: first derivatives are equal
* C²: first and second derivatives are equal
* C^"n": first through "n"^th derivatives are equal

A curve with "C"^"n" parametric continuity also has "G"^"n" geometric continuity. Geometric continuity describes the shape of a curve or surface; parametric continuity also describes this, but adds restrictions on the speed with which the parameter traces out the curve.

Applications

Parametric continuity is a common way of determining the precision of curves. For instance, Hermite and Cardinal splines are only C¹ continuous, while B-splines are C² continuous. When constructed correctly, Bézier curves can also be C¹ continuous. [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19900012238_1990012238.pdf]

References