The Affine Distance Symmetry Set
Definitions and Properties
- A sextatic edge of a CEA polygon is an edge whose affine curvature is a local extremum.
- Each edge of the Affine Evolute can be oriented in the direction of its corresponding Affine Normal Vector. If two edges point to (or away from) their mutual vertex, that vertex is a cusp of the Affine Evolute.
- The set of all self-intersections of each λ-parallel polygon is the Affine Distance Symmetry Set (ADSS) of the original CEA polygon.
ADSS Applet
In the Geogebra Applet below, you can construct your own CEA octagon (moving P1-P6); the Applet will close the CEAP, calculate the Affine Evolute, and mark in red all sextatic edges and all Affine Evolute cusps.
Then, you can play with the λ-parallel polygons (dragging λ or pressing the “play” button on the lower left corner). Points of the ADSS will be marked blue(ish) as they are found.
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Things to note:
- Each cusp of the Affine Evolute corresponds to a sextatic edge of the original polygon.
- If you keep the points in the original position, the ADSS has 3 branches, which we colored with different hues of blue. You can use the checkboxes to see them one at a time.
- In general, the ADSS can always be separated in continuous branches; all of them start and end at cusps of the affine evolute.
- In general, CEA polygons must have at least 6 sextatic edges, so the ADSS must have at least 3 branches.
- If you want to move λ very slowly, you can click on the λ-bar and use the left and right arrows.
- If you want to get rid of blue lines the Applet has drawn, a zoom-in or zoom-out (use mouse wheel) will quickly do so.
Reference
[2] M. Craizer, R. Teixeira and M. Horta, “Affine properties of convex equal-area polygons”; Discrete & Computational Geometry, October 2012, Volume 48, Issue 3, pp 580-595.
