# Parallel curves to a parabola

Topic:
Parabola

## Notes about the parallel curves to a parabola

It never occurred to me before (or I never thought about it) that the parallel curve to a parabola it's not a parabola! By parallel curve I mean a curve that keeps an equal distance along the normal line to the parabola at all its points. Following my curiosity about the parallel curves of a parabola I found out that there are some literature on the web about them. In general the parallel curves (also called offset curves in the CAD area) to some given curve have a more complicated mathematical structure than the progenitor. In the case of a parabola its parallel curves might even be self-intersecting curves, and this implies they must have a higher order algebraic equation. A parabola of equation can be rewritten in parametric form as With some trigonometry we can derive the parametric equations of the curve parallel to the parabola at a distance : where is the angle between the tangent line to the parabola at the point and the x axis. Since is related to the parameter t by it will also be The parallel curve becomes a self-intersecting curve for For the records the Cartesian implicit equation (comprising both the interior and the exterior parallels at a distance ) is: Much better stick to its parametric form. References Wikipedia - Parallel curve Math.StackExchange: Formula for curve parallel to a parabola F. Max Stein: The Curve Parallel to a Parabola is not a Parabola: Parallel Curves