Equidistants

Definition

Given a CPOS polygon P₁P₂…P_2n and a real number λ, we define its λ-equidistant polygon P(λ) by the vertices which split the great diagonals in the ratio λ:(1- λ). More explicitly, the vertices of P(λ) are

P_i(λ)=P_i + λ(P_n+i-P_i),   i=1,2,…,2n

It is easy to see that:

• The 0-equidistant is the original polygon (and so is the 1-equidistant);
• Actually, the λ-equidistant coincides with the (1- λ)-equidistant, just reordered;
• The 5-equidistant is the Area Evolute (AE), traveled twice;
• All equidistants are POSPs. Not all of them are convex, but there must be a neighborhood of λ=0 which produces convex equidistants;
• All (convex) equidistants have the same Central Symmetry Set (CSS) and the same Area Evolute (AE).

Cusps of the equidistants and the CSS

When the λ-triangle P_i-1P_iP_i+1 has a different orientation than the original P_i-1P_iP_i+1, we say that P_i(λ) is a cusp of the λ-equidistant.

The 0.2-equidistant of a CPOS polygon (pink). Labeled green vertices are cusps.

In this next Geogebra Applet, you can construct your 10-sided CPOSP as before (moving P₁-P₉) and play with the λ-equidistants (drag the λ-slider or press the “play” button on the lower left corner; you can use the mouse wheel to zoom in or out AND erase previous cusp traces, if you wish). Whenever a vertex of the equidistant turns into a cusp, we mark it green. As you move λ, what do they trace?

     This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com

In fact (see [1] for proofs):

·        Consecutive edges of the equidistant form a cusp if, and only if, such edges do not cross the corresponding great diagonal;

·        The cusps of the equidistants form the Central Symmetry Set of the original CPOSP.

Reference

[1] M. Craizer, R. Teixeira and M. Horta, “Parallel Opposite Sides Polygons”, preprint.