Editing Parallel curve (section)

===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>.