- Parametric continuity
Parametric continuity is a concept applied to
parametric curve s 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 equalA 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 thespeed 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
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