A Double Parabola with Retroreflection Properties

This Demonstration shows the reflective behavior of a two-dimensional shape formed by similar arcs of two parabolas. An interesting and unexpected reflective behavior arises when this shape takes a specific configuration, with two parabolas of unit focal length. In this particular configuration it is found that, except for the rays with absolute value of the entry angle less than , a ray emerges from the cavity with a trajectory that is nearly opposite to its entry trajectory. This characterizes a cavity with retroreflective behavior. Even in the case of trajectories with entry angle below (shown as dashed trajectories), the exit direction does not appear to vary greatly from the entry direction.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The two sections of the curve considered here are similar arcs of two parabolas with the common vertical axis coincident with the line of entry of the cavity and concavities turned toward one another. The optimal configuration can only be achieved when the focus of each one coincides with the vertex of the other (the focal length equals 1).
The authors [1] found this optimal configuration searching for shapes of bodies that maximize Newton’s aerodynamic resistance. But its application does not lie solely in maximization resistance; it can be used in the design of retroreflectors. As can be verified here, the double parabola shape, although it does not guarantee perfect inversion of all the incident radiation, fulfills this function with great success: it gives a good approximation for a significant fraction of the entry angles, and even for the rest, it does not give too great an error. This is very promising for new geometries for optical elements that make up retroreflecting surfaces. See [2] for applications to the automobile industry, including road signs.
[1] P. D. F. Gouveia, A. Plakhov, and D. F. M. Torres, "Two-Dimensional Body of Maximum Mean Resistance," Applied Mathematics and Computation 215, 2009 pp. 37–52.
[2] P. D. F. Gouveia, "Computation of Variational Symmetries and Optimization of Newtonian Aerodynamic Resistance" (in Portuguese), Ph.D. thesis, University of Aveiro, Portugal, February 2008.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+