Editing Parallel curve (section)

====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]].