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 (Y, V) similarly to Nomizu and Sasaki (Nagoya Math J 132:63–90, 1993). We prove that vertices of (X, U) correspond to flattening points for (Y, V) 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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Arnold, V.: On the number of flattening points of space curves. In: Bunimovich, L.A., Gurevich, B.M., Pesin, Ya.B. (eds.) Sinai’s Moscow Seminar on Dynamical Systems. American Mathematical Society Translations, vol. 171, pp. 11–22. American Mathematical Society, Providence (1996)
Craizer, M., Garcia, R.A.: Umbilical centroaffine codimension 2 immersions and Loewner’s type conjectures. (2018). arXiv:1811.07331
Craizer, M., Pesco, S.: Affine geometry of equal-volume polygons in \(3\)-space. Comput. Aided Geom. Des. 57, 44–56 (2017)
Craizer, M., Teixeira, R.C., da Silva, M.A.H.B.: Affine properties of convex equal-area polygons. Discrete Comput. Geom. 48(3), 580–595 (2012)
Nomizu, K., Sasaki, T.: Centroaffine immersions of codimension two and projective hypersurface theory. Nagoya Math. J. 132, 63–90 (1993)
Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge Tracts in Mathematics, vol. 111. Cambridge University Press, Cambridge (1994)
Pak, I.: Lectures on Discrete and Polyhedral Geometry (2010). http://www.math.ucla.edu/~pak/book.htm
Tabachnikov, S.: A four vertex theorem for polygons. Am. Math. Monthly 107(9), 830–833 (2000)
Uribe-Vargas, R.: On \(4\)-flattening theorems and the curves of Carathéodory, Barner and Segre. J. Geom. 77(1–2), 184–192 (2003)
The authors want to thank CNPq and CAPES for financial support during the preparation of this manuscript.
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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
- Affine vertices
- Planar points
- Flattening points
- Support points
- Four vertices theorem
- Four planar points theorem
Mathematics Subject Classification