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Centroaffine Duality for Spatial Polygons

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Abstract

We discuss centroaffine geometry of polygons in 3-space. For a polygon X that is locally convex with respect to an origin together with a transversal vector field U, we define the centroaffine dual pair (YV) similarly to Nomizu and Sasaki (Nagoya Math J 132:63–90, 1993). We prove that vertices of (XU) correspond to flattening points for (YV) and also that constant curvature polygons are dual to planar polygons. As an application, we give a new proof of a known four flattening points theorem for spatial polygons.

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Acknowledgements

The authors want to thank CNPq and CAPES for financial support during the preparation of this manuscript.

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Correspondence to Marcos Craizer.

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Craizer, M., Pesco, S. Centroaffine Duality for Spatial Polygons. Discrete Comput Geom (2019). https://doi.org/10.1007/s00454-019-00136-4

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Keywords

  • Affine vertices
  • Planar points
  • Flattening points
  • Support points
  • Four vertices theorem
  • Four planar points theorem

Mathematics Subject Classification

  • 53A15
  • 52C35