Parametric continuity

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

Contents 1 Definition 2 Order of continuity 3 Applications 4 References

[edit] Definition

A curve can be said to have Cⁿ continuity if

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 C¹ 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.

[edit] Order of continuity

The various order of parametric continuity can be described as follows^[1]:

• C⁰: curves are joined
• C¹: first derivatives are equal
• C²: first and second derivatives are equal
• Cⁿ: first through n^th derivatives are equal

A curve with Cⁿ parametric continuity also has Gⁿ 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.

[edit] 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.^[2]

[edit] References

1. ^ Parametric Curves
2. ^ http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19900012238_1990012238.pdf