Editing Parallel curve (section)

===Geometric properties:<ref name="hart30">E. Hartmann: [http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf ''Geometry and Algorithms for COMPUTER AIDED DESIGN.''] S. 30.</ref>===
*<math>\vec x'_d(t) \parallel  \vec x'(t),\quad</math> that means: the tangent vectors for a fixed parameter are parallel.
*<math>k_d(t)=\frac{k(t)}{1+dk(t)},\quad</math> with <math>k(t)</math> the  [[curvature]] of the given curve and <math>k_d(t)</math> the curvature of the parallel curve for parameter <math>t</math>.
*<math>R_d(t)=R(t) + d,\quad</math> with <math>R(t)</math> the  [[curvature#Curvature of plane curves|radius of curvature]] of the given curve and <math>R_d(t)</math> the radius of curvature of the parallel curve for parameter <math>t</math>.
* When they exist, the [[Osculating circle|osculating circles]] to parallel curves at corresponding points are concentric. <ref>Fiona O'Neill: [https://fionasmathblog.com/2022/04/26/planar-bertrand-curves-with-pictures/ ''Planar Bertrand Curves (with Pictures!).'']</ref>
*As for [[parallel (geometry)|parallel lines]], a normal line to a curve is also normal to its parallels.
*When parallel curves are constructed they will have [[Cusp (singularity)|cusp]]s when the distance from the curve matches the radius of [[curvature]]. These are the points where the curve touches the [[evolute]].
*If the progenitor curve is a boundary of a planar set and its parallel curve is without self-intersections, then the latter is the boundary of the [[Minkowski sum]] of the planar set and the disk of the given radius.

If the given curve is polynomial (meaning that <math>x(t)</math> and <math>y(t)</math> are polynomials), then the parallel curves are usually not polynomial. In CAD area this is a drawback, because CAD systems use polynomials or rational curves. In order to get at least rational curves, the square root of the representation of the parallel curve has to be solvable. Such curves are called ''pythagorean hodograph curves'' and were investigated by R.T. Farouki.<ref>Rida T. Farouki:
''Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable (Geometry and Computing).'' Springer, 2008, {{ISBN|978-3-540-73397-3}}.</ref>