Editing Parallel curve (section)

===Normal fans===
As described [[#Parallel curve of a parametrically given curve|above]], the parametric representation of a parallel curve, <math>\vec x_d(t)</math>, to a given curver, <math>\vec x(t)</math>, with distance <math>|d|</math> is:

:<math>\vec x_d(t) = \vec x(t) + d\vec n(t)</math> with the unit normal <math>\vec n(t)</math>.

At a sharp corner (<math>t = t_c</math>), the normal to <math>\vec x(t_c)</math> given by <math>\vec n(t_c)</math> is discontinuous, meaning the [[one-sided limit]] of the normal from the left <math>\vec n(t_c^-)</math> is unequal to the limit from the right <math>\vec n(t_c^+)</math>. Mathematically,
:<math>\vec n(t_c^-) = \lim_{t \to t_c^-}\vec n(t) \ne \vec n(t_c^+) = \lim_{t \to t_c^+}\vec n(t)</math>.

[[File:Normal fan for defining parallel curves.png|thumb|Normal fan for defining parallel curves around a sharp corner]]
However, we can define a normal fan<ref name="Brechner1990"/> <math>\vec n_f(\alpha)</math> that provides an [[Interpolation|interpolant]] between <math>\vec n(t_c^-)</math> and <math>\vec n(t_c^+)</math>, and use <math>\vec n_f(\alpha)</math> in place of <math>\vec n(t_c)</math> at the sharp corner:
:<math>\vec n_f(\alpha) = \frac{(1 - \alpha)\vec n(t_c^-) + \alpha\vec n(t_c^+)}{\lVert (1 - \alpha)\vec n(t_c^-) + \alpha\vec n(t_c^+) \rVert},\quad</math>where <math>0 < \alpha < 1</math>.

The resulting definition of the parallel curve <math>\vec x_d(t)</math> provides the desired behavior:
:<math>\vec x_d(t) = \begin{cases}
\vec x(t) + d\vec n(t), & \text{if }t < t_c\text{ or }t > t_c \\
\vec x(t_c) + d\vec n_f(\alpha), & \text{if }t = t_c\text{ where }0 < \alpha < 1
\end{cases}</math>