properties of parallel curves

• Two plane curves are parallel curves of each other, if every normal of one curve is also a normal of the other curve (then one may show that the distance of the corresponding points of the curves is a ).

• Two curves are parallel curves of each other, if they are the loci of the end points of a line segment which moves perpendicularly to its own direction.

• Every regular curve having a continuous curvature has an infinite family of parallel curves.

• The parallelism of curves is an equivalence relation.

• The two parallel curves ${\gamma }_{\pm a}$ on both sides of a curve $\gamma$ at the distance $a$ form the envelope of the family of circles with center on $\gamma$ and radius $a$.

• If $\gamma$ is a and closed curve with perimeter $p$, then the perimeter of ${\gamma }_{\pm a}$ is equal to  $p\pm 2\pi a$  and the area between $\gamma$ and the parallel curve is equal to $pa\pm \pi a^2$ (one must also assume that the parallel curve don’t intersect the evolute of $\gamma$).