Area Evolute and Central Symmetry Set

Given a convex parallel opposite sides polygon (CPOSP) P₁P₂…P_2n we can define:

• Its Great Diagonals: segments d_i that join P_i and P_i+n (i=1,2,…,n).
• ItsArea Evolute (AE): polygon whose vertices M_i are the midpoints of the Great Diagonals.
• Its Central Symmetry Set (CSS): polygon whose vertices D_i+½ are intersections of successive great diagonals.

Note that:

• The AE reduces to a point if, and only if, the polygon is symmetric with relation to that point (by definition);
• The CSS of a CPOSP reduces to a point if, and only if, the polygon is symmetric with relation to that point.

Cusps of the AE and the CSS

• A cusp of the CSS is a vertex D_i+½ such that the orientations of the triangles D_i-1+½ D_i+½ D_i+1+½ and P_i-1P_iP_i+1 are reversed.
• A cusp of the AE is a vertex M_i such that the orientations of the triangles M_i-1M_iM_i+1 and P_i-1P_iP_i+1 are reversed (equivalently: M_i is a cusp if M_i-1 and M_i+1 are on the same side of d_i).

In this Geogebra applet, the vertices of the AE and of the CSS are marked red whenever they turn into cusps (as you play with the outer 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

For non-degenerate positions (see [1] for proofs):

• The number of cusps in the CSS is odd and at least 3;
• The number of cusps in the AE is odd and at least 3;
• A vertex of the AE is a cusp if, and only if, it belongs to the CSS.

Alternate CSS cusp definition

If we define λ_i+½ (i=1,2,…,2n) by the equations

D_i+½ = P_i + λ_i+½·(P_i+n – P_i) = P_i+1 + λ_i+½·(P_i+1+n – P_i+1)

then we have

D_i+½ – D_i-1+½ = (λ_i+½ – λ_i-1+½)(P_i+n – P_i)

Now, in a CPOSP, the great diagonals P_i+n – P_i “turn” in the same orientation as the polygon itself. Therefore the triangles D_i-1+½ D_i+½ D_i+1+½ and P_i-1P_iP_i+1 have opposite orientations if and only if

(λ_i+½ – λ_i-1+½)(λ_i+1+½ – λ_i+½)<0

In short

D_i+½ is a cusp of the CSS if, and only if, λ_i+½ is a local extremum of the cyclic sequence {λ_i+½}

References

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