# Parallel curves to a parabola

- Author:
- Luca Moroni

- Topic:
- Parabola

## Parallel curves to a parabola

## Notes about the parallel curves to a parabola

It never occurred to me before (or I never thought about it) that 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 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.

*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***the parallel curve to a parabola it's not a parabola!***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*t*by**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