Editing Smoothness (section)

===Parametric continuity===
'''Parametric continuity''' ('''''C'''''<sup>'''''k'''''</sup>) is a concept applied to [[parametric curve]]s, which describes the smoothness of the parameter's value with distance along the curve. A (parametric) curve <math>s:[0,1]\to\mathbb{R}^n</math> is said to be of class ''C''<sup>''k''</sup>, if <math>\textstyle \frac{d^ks}{dt^k}</math> exists and is continuous on <math>[0,1]</math>, where derivatives at the end-points <math>0</math> and <math>1</math> are taken to be [[Semi-differentiability|one sided derivatives]] (from the right at <math>0</math> and from the left at <math>1</math>).

As a practical application of this concept, a curve describing the motion of an object with a parameter of time must have ''C''<sup>1</sup> continuity and its first derivative is differentiable—for the object to have finite acceleration. For smoother motion, such as that of a camera's path while making a film, higher orders of parametric continuity are required.

====Order of parametric continuity====
[[File:Parametric continuity C0.svg|upright=1.2|thumb|Two [[Bézier curve]] segments attached that is only C<sup>0</sup> continuous]]
[[File:Parametric continuity vector.svg|upright=1.2|thumb|Two Bézier curve segments attached in such a way that they are C<sup>1</sup> continuous]]
The various order of parametric continuity can be described as follows:<ref>{{cite web |first=Michiel |last=van de Panne |url=https://www.cs.helsinki.fi/group/goa/mallinnus/curves/curves.html |title=Parametric Curves |work=Fall 1996 Online Notes |date=1996 |publisher=University of Toronto, Canada |access-date=2019-09-01 |archive-date=2020-11-26 |archive-url=https://web.archive.org/web/20201126212511/https://www.cs.helsinki.fi/group/goa/mallinnus/curves/curves.html |url-status=live }}</ref>
* <math>C^0</math>: zeroth derivative is continuous (curves are continuous)
* <math>C^1</math>: zeroth and first derivatives are continuous
* <math>C^2</math>: zeroth, first and second derivatives are continuous
* <math>C^n</math>: 0-th through <math>n</math>-th derivatives are continuous