See planetmath.
Given two curves, one is a parallel curve (also known as an offset curve) of the other if the points on the first curve are equidistant to the corresponding points in the direction of the second curve’s normal. Alternatively, a parallel of a curve can be defined as the envelope of congruent circles whose centres lie on the curve.
For a parametric curve in the plane defined by $F(u):=(x(u),y(u))$, its parallel curve $G(u):=(X(u),Y(u))$ with offset $t$ is defined by
$$ \begin{array}{l} X(u)=x(u)+\frac{t y^{\prime}(u)}{\sqrt{x^{\prime}(u)^{2}+y^{\prime}(u)^{2}}} \\ Y(u)=y(u)-\frac{t x^{\prime}(u)}{\sqrt{x^{\prime}(u)^{2}+y^{\prime}(u)^{2}}} \end{array} $$________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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