Editing Parallel curve (section)

== Generalizations ==
The problem generalizes fairly obviously to higher dimensions e.g. to offset surfaces, and slightly less trivially to [[pipe surface]]s.<ref name="PottmannWallner2001">{{cite book|first1=Helmut|last1=Pottmann|first2=Johannes|last2=Wallner|title=Computational Line Geometry|url=https://books.google.com/books?id=6ZrqcYKgtE0C&pg=PA303|year=2001|publisher=Springer Science & Business Media|isbn=978-3-540-42058-3|pages=303–304}}</ref> Note that the terminology for the higher-dimensional versions varies even more widely than in the planar case, e.g. other authors speak of parallel fibers, ribbons, and tubes.<ref name="Chirikjian2009">{{cite book|author1-link=Gregory S. Chirikjian|first=Gregory S.|last=Chirikjian|title=Stochastic Models, Information Theory, and Lie Groups, Volume 1: Classical Results and Geometric Methods|year=2009|publisher=Springer Science & Business Media|isbn=978-0-8176-4803-9|pages=171–175}}</ref> For curves embedded in 3D surfaces the offset may be taken along a [[geodesic]].<ref name="Sarfraz2003">{{cite book|editor-first=Muhammad|editor-last=Sarfraz|title=Advances in geometric modeling|url=https://books.google.com/books?id=kfZQAAAAMAAJ&pg=PA72|year=2003|publisher=Wiley|isbn=978-0-470-85937-7|page=72}}</ref>

Another way to generalize it is (even in 2D) to consider a variable distance, e.g. parametrized by another curve.<ref name="barn"/> One can for example stroke (envelope) with an ellipse instead of circle<ref name="barn"/> as it is possible for example in [[METAFONT]].<ref>{{Cite journal |last=Kinch |first=Richard J. |date=1995 |title=MetaFog: Converting METAFONT Shapes to Contours |url=https://www.tug.org/TUGboat/tb16-3/tb48kinc.pdf |journal=TUGboat |volume=16 |issue=3 |pages=233-243}}</ref>
[[File:Envelope of ellipses.png|thumb|An envelope of ellipses forming two general offset curves above and below a given curve]]
More recently [[Adobe Illustrator]] has added somewhat similar facility in version [[CS5]], although the control points for the variable width are visually specified.<ref>http://design.tutsplus.com/tutorials/illustrator-cs5-variable-width-stroke-tool-perfect-for-making-tribal-designs--vector-4346 application of the generalized version in Adobe Illustrator CS5 (also [http://tv.adobe.com/watch/learn-illustrator-cs5/using-variablewidth-strokes/ video])</ref> In contexts where it's important to distinguish between constant and variable distance offsetting the acronyms CDO and VDO are sometimes used.<ref name="jarek"/>

===General offset curves===
Assume you have a regular parametric representation of a curve, <math> \vec x(t) = (x(t),y(t))</math>, and you have a second curve that can be parameterized by its unit normal, <math> \vec d(\vec n)</math>, where the normal of <math>\vec d(\vec n) = \vec n</math> (this parameterization by normal exists for curves whose curvature is strictly positive or negative, and thus convex, smooth, and not straight). The parametric representation of the general offset curve of <math>\vec x(t)</math> offset by <math> \vec d(\vec n)</math> is:

:<math> \vec x_d(t)=\vec x(t)+ \vec d(\vec n(t)), \quad</math> where <math>\vec n(t)</math> is the unit normal of <math>\vec x(t)</math>.
Note that the trival offset, <math>\vec d(\vec n) = d\vec n</math>, gives you ordinary parallel (aka, offset) curves.

====Geometric properties:<ref name="barn"/>====
*<math>\vec x'_d(t) \parallel  \vec x'(t),\quad</math> that means: the tangent vectors for a fixed parameter are parallel.
*As for [[parallel (geometry)|parallel lines]], a normal to a curve is also normal to its general offsets.
*<math>k_d(t)=\dfrac{k(t)}{1+\dfrac{k(t)}{k_n(t)}},\quad</math> with <math>k_d(t)</math> the  [[curvature]] of the general offset curve, <math>k(t)</math> the curvature of <math>\vec x(t)</math>, and <math>k_n(t)</math> the curvature of <math>\vec d(\vec n(t))</math> for parameter <math>t</math>.
*<math>R_d(t)=R(t) + R_n(t),\quad</math> with <math>R_d(t)</math> the  [[curvature#Curvature of plane curves|radius of curvature]] of the general offset curve, <math>R(t)</math> the radius of curvature of <math>\vec x(t)</math>, and <math>R_n(t)</math> the radius of curvature of <math>\vec d(\vec n(t))</math> for parameter <math>t</math>.
*When general offset curves are constructed they will have [[Cusp (singularity)|cusp]]s when the [[curvature]] of the curve matches curvature of the offset. These are the points where the curve touches the [[evolute]].

===General offset surfaces===
General offset surfaces describe the shape of cuts made by a variety of cutting bits used by three-axis end mills in [[numerically controlled]] [[machining]].<ref name="Brechner1990"/> Assume you have a regular parametric representation of a surface, <math> \vec x(u,v) = (x(u,v),y(u,v),z(u,v))</math>, and you have a second surface that can be parameterized by its unit normal, <math> \vec d(\vec n)</math>, where the normal of <math>\vec d(\vec n) = \vec n</math> (this parameterization by normal exists for surfaces whose [[Gaussian curvature]] is strictly positive, and thus convex, smooth, and not flat). The parametric representation of the general offset surface of <math>\vec x(t)</math> offset by <math> \vec d(\vec n)</math> is:

:<math> \vec x_d(u,v)=\vec x(u,v)+ \vec d(\vec n(u,v)), \quad</math> where <math>\vec n(u,v)</math> is the unit normal of <math>\vec x(u,v)</math>.
Note that the trival offset, <math>\vec d(\vec n) = d\vec n</math>, gives you ordinary parallel (aka, offset) surfaces.

====Geometric properties:<ref name="barn"/>====
*As for [[parallel (geometry)|parallel lines]], the tangent plane of a surface is parallel to the tangent plane of its general offsets.
*As for [[parallel (geometry)|parallel lines]], a normal to a surface is also normal to its general offsets.
*<math>S_d = (1 + SS_n^{-1})^{-1} S, \quad</math> where <math>S_d, S,</math> and <math>S_n</math> are the [[shape operator]]s for <math>\vec x_d, \vec x,</math> and <math>\vec d(\vec n)</math>, respectively.
:The principal curvatures are the [[Eigenvalues and eigenvectors|eigenvalues]] of the [[shape operator]], the principal curvature directions are its [[Eigenvalues and eigenvectors|eigenvectors]], the [[Gaussian curvature]] is its [[determinant]], and the mean curvature is half its [[trace (linear algebra)|trace]].
*<math>S_d^{-1} = S^{-1} + S_n^{-1}, \quad</math> where <math>S_d^{-1}, S^{-1}</math> and <math>S_n^{-1}</math> are the inverses of the [[shape operator]]s for <math>\vec x_d, \vec x,</math> and <math>\vec d(\vec n)</math>, respectively.
:The principal radii of curvature are the [[Eigenvalues and eigenvectors|eigenvalues]] of the inverse of the [[shape operator]], the principal curvature directions are its [[Eigenvalues and eigenvectors|eigenvectors]], the reciprocal of the [[Gaussian curvature]] is its [[determinant]], and the mean radius of curvature is half its [[trace (linear algebra)|trace]].
Note the similarity to the geometric properties of [[#General offset curves|general offset curves]].

===Derivation of geometric properties for general offsets===
The geometric properties listed above for general offset curves and surfaces can be derived for offsets of arbitrary dimension. Assume you have a regular parametric representation of an n-dimensional surface, <math> \vec x(\vec u)</math>, where the dimension of <math>\vec u</math> is n-1. Also assume you have a second n-dimensional surface that can be parameterized by its unit normal, <math> \vec d(\vec n)</math>, where the normal of <math>\vec d(\vec n) = \vec n</math> (this parameterization by normal exists for surfaces whose [[Gaussian curvature]] is strictly positive, and thus convex, smooth, and not flat). The parametric representation of the general offset surface of <math>\vec x(\vec u)</math> offset by <math> \vec d(\vec n)</math> is:

:<math> \vec x_d(\vec u) = \vec x(\vec u)+ \vec d(\vec n(\vec u)), \quad</math> where <math>\vec n(\vec u)</math> is the unit normal of <math>\vec x(\vec u)</math>. (The trival offset, <math>\vec d(\vec n) = d\vec n</math>, gives you ordinary parallel surfaces.)

First, notice that the normal of <math>\vec x(\vec u) = </math> the normal of <math>\vec d(\vec n(\vec u)) = \vec n(\vec u),</math> by definition. Now, we'll apply the differential w.r.t. <math>\vec u</math> to <math>\vec x_d</math>, which gives us its tangent vectors spanning its tangent plane.
:<math> \partial\vec x_d(\vec u) = \partial\vec x(\vec u)+ \partial\vec d(\vec n(\vec u))</math>
Notice, the tangent vectors for <math>\vec x_d</math> are the sum of tangent vectors for <math>\vec x(\vec u)</math> and its offset <math> \vec d(\vec n)</math>, which share the same unit normal. Thus, '''the general offset surface shares the same tangent plane and normal with''' <math>\vec x(\vec u)</math> and <math>\vec d(\vec n(\vec u))</math>. That aligns with the nature of envelopes.

We now consider the [[Weingarten equations]] for the [[shape operator]], which can be written as <math>\partial\vec n = -\partial\vec xS</math>. If <math>S</math> is invertable, <math>\partial\vec x = -\partial\vec nS^{-1}</math>. Recall that the principal curvatures of a surface are the [[Eigenvalues and eigenvectors|eigenvalues]] of the shape operator, the principal curvature directions are its [[Eigenvalues and eigenvectors|eigenvectors]], the Gauss curvature is its [[determinant]], and the mean curvature is half its [[trace (linear algebra)|trace]]. The inverse of the shape operator holds these same values for the radii of curvature.

Substituting into the equation for the differential of <math>\vec x_d</math>, we get:
:<math> \partial\vec x_d = \partial\vec x - \partial\vec n S_n^{-1},\quad</math> where <math>S_n</math> is the shape operator for <math>\vec d(\vec n(\vec u))</math>.

Next, we use the [[Weingarten equations]] again to replace <math>\partial\vec n</math>:
:<math>\partial\vec x_d = \partial\vec x + \partial\vec x S S_n^{-1},\quad</math> where <math>S</math> is the shape operator for <math>\vec x(\vec u)</math>.

Then, we solve for <math>\partial\vec x</math> and multiple both sides by <math>-S</math> to get back to the [[Weingarten equations]], this time for <math>\partial\vec x_d</math>:
:<math>\partial\vec x_d (I + S S_n^{-1})^{-1} = \partial\vec x,</math>
:<math>-\partial\vec x_d (I + S S_n^{-1})^{-1}S = -\partial\vec xS = \partial\vec n.</math>

Thus, <math>S_d = (I + S S_n^{-1})^{-1}S</math>, and inverting both sides gives us, <math>S_d^{-1} = S^{-1} + S_n^{-1}</math>.