Editing Parallel curve (section)

== Parallel (offset) surfaces ==
[[File:Offset surface.png|thumb|right|Offset surface of a complex irregular shape]]
Offset surfaces are important in [[numerically controlled]] [[machining]], where they describe the shape of the cut made by a ball nose end mill of a three-axis mill.<ref name="Faux1979"/> If there is a regular parametric representation <math> \vec x(u,v) = (x(u,v),y(u,v),z(u,v))</math> of the given surface available, the second definition of a parallel curve (see above) generalizes to the following parametric representation of the parallel surface with distance <math> |d| </math>:

:<math> \vec x_d(u,v)=\vec x(u,v)+d\vec n(u,v)</math> with the unit normal <math>\vec n_d(u,v) = {{{\partial \vec x \over \partial u} \times {\partial \vec x \over \partial v}} \over {|{{\partial \vec x \over \partial u} \times {\partial \vec x \over \partial v}}|}}</math>.

Distance parameter <math>d</math> may be negative, too. In this case one gets a parallel surface on the opposite side of the surface (see similar diagram on the parallel curves of a circle). One easily checks: a parallel surface of a plane is a parallel plane in the common sense and the parallel surface of a sphere is a concentric sphere.

===Geometric properties:<ref name="barn">{{cite book|editor-first=Robert E.|editor-last=Barnhill|title=Geometry Processing for Design and Manufacturing|year=1992|publisher=SIAM|isbn=978-0-89871-280-3|first=Eric L.|last=Brechner|chapter=5. General Offset Curves and Surfaces|pages=101–}}</ref>===
*<math>{\partial \vec x_d \over \partial u} \parallel  {\partial \vec x \over \partial u}, \quad {\partial \vec x_d \over \partial v} \parallel  {\partial \vec x \over \partial v}, \quad</math> that means: the tangent vectors for fixed parameters are parallel.
*<math>\vec n_d(u,v) = \pm\vec n(u,v), \quad</math> that means: the normal vectors for fixed parameters match direction.
*<math>S_d = (1 + d S)^{-1} S, \quad</math> where <math>S_d</math> and <math>S</math> are the [[shape operator]]s for <math>\vec x_d</math> and <math>\vec x</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} + d I, \quad</math> where <math>S_d^{-1}</math> and <math>S^{-1}</math> are the inverses of the [[shape operator]]s for <math>\vec x_d</math> and <math>\vec x</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 [[#Parallel curve of a parametrically given curve|parallel curves]].