Editing Smoothness (section)

==Continuity==

The terms ''parametric continuity'' (''C''<sup>''k''</sup>) and ''geometric continuity'' (''G<sup>n</sup>'') were introduced by [[Brian A. Barsky|Brian Barsky]], to show that the smoothness of a curve could be measured by removing restrictions on the [[speed]], with which the parameter traces out the curve.<ref name="Barsky1981">{{cite thesis |type=Ph.D. |last=Barsky |first=Brian A. |date=1981 |title=The Beta-spline: A Local Representation Based on Shape Parameters and Fundamental Geometric Measures |publisher=University of Utah, Salt Lake City, Utah|url=https://dl.acm.org/citation.cfm?id=910231 }}</ref><ref name="Barsky1988">{{cite book|author=Brian A. Barsky|title=Computer Graphics and Geometric Modeling Using Beta-splines|year=1988|publisher=Springer-Verlag, Heidelberg|isbn=978-3-642-72294-3}}</ref><ref name="BartelsBeattyBarsky1987">{{cite book|author1=Richard H. Bartels|author2=John C. Beatty|author3=Brian A. Barsky|title=An Introduction to Splines for Use in Computer Graphics and Geometric Modeling|year=1987|publisher=Morgan Kaufmann|isbn=978-1-55860-400-1|at=Chapter 13. Parametric vs. Geometric Continuity}}</ref>

===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

===Geometric continuity===

{{Distinguish|Geometrical continuity}}
[[File:Curves g1 contact.svg|upright=1.2|thumb|Curves with ''G''<sup>1</sup>-contact (circles,line)]]
[[File:Kegelschnitt-Schar.svg|upright=1.2|thumb|<math>(1-\varepsilon^2) x^2 -2px+y^2=0 , \ p>0 \ , \varepsilon\ge 0</math><br />
pencil of conic sections with ''G''<sup>2</sup>-contact: p fix, <math>\varepsilon</math> variable <br />
(<math>\varepsilon=0</math>: circle,<math>\varepsilon=0.8</math>: ellipse, <math>\varepsilon=1</math>: parabola, <math>\varepsilon=1.2</math>: hyperbola)]]

A [[curve]] or [[Surface (topology)|surface]] can be described as having <math>G^n</math> continuity, with <math>n</math> being the increasing measure of smoothness. Consider the segments either side of a point on a curve:

*<math>G^0</math>: The curves touch at the join point.
*<math>G^1</math>: The curves also share a common [[tangent]] direction at the join point.
*<math>G^2</math>: The curves also share a common center of curvature at the join point.

In general, <math>G^n</math> continuity exists if the curves can be reparameterized to have <math>C^n</math> (parametric) continuity.<ref name=Barsky-DeRose>{{cite journal |first1=Brian A. |last1=Barsky |first2=Tony D. |last2=DeRose |title=Geometric Continuity of Parametric Curves: Three Equivalent Characterizations |journal=IEEE Computer Graphics and Applications |volume=9 |issue=6 |year=1989 |pages=60–68 |doi=10.1109/38.41470 |s2cid=17893586 }}</ref><ref>{{cite web |url=https://www2.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf#page=55 |first=Erich |last=Hartmann |title=Geometry and Algorithms for Computer Aided Design |page=55 |date=2003 |publisher=[[Technische Universität Darmstadt]] |access-date=2019-08-31 |archive-date=2020-10-23 |archive-url=https://web.archive.org/web/20201023054532/http://www2.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf#page=55 |url-status=live }}</ref> A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.

Equivalently, two vector functions <math>f(t)</math> and <math>g(t)</math> such that <math>f(1)=g(0)</math> have <math>G^n</math> continuity at the point where they meet if
they satisfy equations known as Beta-constraints.  For example, the Beta-constraints for <math>G^4</math> continuity are:

:<math>
\begin{align}
g^{(1)}(0) & = \beta_1 f^{(1)}(1) \\
g^{(2)}(0) & = \beta_1^2 f^{(2)}(1) + \beta_2 f^{(1)}(1) \\
g^{(3)}(0) & = \beta_1^3 f^{(3)}(1) + 3\beta_1\beta_2 f^{(2)}(1) +\beta_3 f^{(1)}(1) \\
g^{(4)}(0) & = \beta_1^4 f^{(4)}(1) + 6\beta_1^2\beta_2 f^{(3)}(1) +(4\beta_1\beta_3+3\beta_2^2) f^{(2)}(1) +\beta_4 f^{(1)}(1) \\
\end{align}
</math>
where <math>\beta_2</math>, <math>\beta_3</math>, and <math>\beta_4</math> are arbitrary, but <math>\beta_1</math> is constrained to be positive.{{r|Barsky-DeRose|p=65}}
In the case <math>n=1</math>, this reduces to 
<math>f'(1)\neq0</math> and <math>f'(1) =  kg'(0)</math>, for a scalar <math>k>0</math> (i.e., the direction, but not necessarily the magnitude, of the two vectors is equal).

While it may be obvious that a curve would require <math>G^1</math> continuity to appear smooth, for good [[aesthetics]], such as those aspired to in [[architecture]] and [[sports car]] design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has <math>G^2</math> continuity.{{cn|date=April 2024}}

A {{em|[[rounded rectangle]]}} (with ninety degree circular arcs at the four corners) has <math>G^1</math> continuity, but does not have <math>G^2</math> continuity. The same is true for a {{em|[[rounded cube]]}}, with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with <math>G^2</math> continuity is required, then [[cubic splines]] are typically chosen; these curves are frequently used in [[industrial design]].