Parallel Opposite Sides Polygons (POSP)

Definition

As the name indicates, a plane polygon with 2n vertices P₁P₂…P_2n is called Parallel Opposite Sides when P_iP_i+1 is parallel to P_i+nP_i+n+1 for i=1,2,…,n (indices are taken modulo 2n, so P_2n+1=P₁). Note that such property is invariant by affine transformations! In these pages, we are interested in Convex POS polygons.

·        n=2: all 4-sided POSPs are parallelograms; they are all equivalent via affine transformations of the plane.

·        n≥3: there are 3n degrees of freedom when defining a 2n-sided POSP (of which, 6 are spent in affine transformations).

Construct your own POS decagon

You can use the Applet below (created with Geogebra) to construct your own 10-sided POS polygon!

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

You can move P₁, P₂, …, P₆ at will, defining the 5 directions of the sides (12 degrees of freedom);
You can move P₇, P₈ and P₉ along the dotted lines (3 more degrees);
Finally, the Applet will determine P₁₀ in order to close a POS. Can you see how this is done?

In the next pages we shall explore some properties of convex POS polygons.

References

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