Editing Parallel curve (section)

===General offset curves===
Assume you have a regular parametric representation of a curve, <math> \vec x(t) = (x(t),y(t))</math>, and you have a second curve 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 curves whose curvature is strictly positive or negative, and thus convex, smooth, and not straight). The parametric representation of the general offset curve of <math>\vec x(t)</math> offset by <math> \vec d(\vec n)</math> is:

:<math> \vec x_d(t)=\vec x(t)+ \vec d(\vec n(t)), \quad</math> where <math>\vec n(t)</math> is the unit normal of <math>\vec x(t)</math>.
Note that the trival offset, <math>\vec d(\vec n) = d\vec n</math>, gives you ordinary parallel (aka, offset) curves.

====Geometric properties:<ref name="barn"/>====
*<math>\vec x'_d(t) \parallel  \vec x'(t),\quad</math> that means: the tangent vectors for a fixed parameter are parallel.
*As for [[parallel (geometry)|parallel lines]], a normal to a curve is also normal to its general offsets.
*<math>k_d(t)=\dfrac{k(t)}{1+\dfrac{k(t)}{k_n(t)}},\quad</math> with <math>k_d(t)</math> the  [[curvature]] of the general offset curve, <math>k(t)</math> the curvature of <math>\vec x(t)</math>, and <math>k_n(t)</math> the curvature of <math>\vec d(\vec n(t))</math> for parameter <math>t</math>.
*<math>R_d(t)=R(t) + R_n(t),\quad</math> with <math>R_d(t)</math> the  [[curvature#Curvature of plane curves|radius of curvature]] of the general offset curve, <math>R(t)</math> the radius of curvature of <math>\vec x(t)</math>, and <math>R_n(t)</math> the radius of curvature of <math>\vec d(\vec n(t))</math> for parameter <math>t</math>.
*When general offset curves are constructed they will have [[Cusp (singularity)|cusp]]s when the [[curvature]] of the curve matches curvature of the offset. These are the points where the curve touches the [[evolute]].