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 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