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.

[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

Two Bézier curve segments attached that is only C⁰ continuous.

Two Bézier curve segments attached in such a way that they are C¹ continuous.

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

The term "parametric continuity" was introduced to distinguish it from "geometric continuity" (Gⁿ) which removes restrictions on the speed with which the parameter traces out the curve (Bartels, Beatty & Barsky 1987, Ch. 13).

[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 cubic B-splines are C² continuous. When constructed correctly, Bézier curves can also be C¹ continuous.^[2]

[edit] References

• Richard H. Bartels, John C Beatty, and Brian A. Barsky (1987). Splines for Use in Computer Graphics and Geometric Modeling. Morgan Kaufmann. ISBN 0-934613-27-3. 

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