Editing Parallel curve (section)

===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]].