PlanetMath: properties of parallel curves

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properties of parallel curves

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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 constant).
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 form the envelope of the family of circles with center on $\gamma$ and radius .
If $\gamma$ is a regular and closed curve with perimeter , 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 $\gamma$ ).