# zbMATH — the first resource for mathematics

Characteristic parameter sets and limits of circulant Hermitian polygon transformations. (English) Zbl 1193.51019
Summary: Polygon transformations based on taking the apices of similar triangles constructed on the sides of an initial polygon are analyzed as well as the limit polygons obtained by iteratively applying such transformations. In contrast to other approaches, this is done with respect to two construction parameters representing a base angle and an apex perpendicular subdivision ratio. Furthermore, a combined transformation leading to circulant Hermitian matrices is proposed, which eliminates the rotational effect of the basic transformation. A finite set of characteristic parameter subdomains is derived for which the sequence converges to specific eigenpolygons. Otherwise, limit polygons turn out to be linear combinations of up to three eigenpolygons. This leads to a full classification of circulant Hermitian similar triangles based polygon transformations and their limit polygons. As a byproduct classical results as Napoleon’s theorem and the Petr-Douglas-Neumann theorem can be easily deduced.

##### MSC:
 51M04 Elementary problems in Euclidean geometries 52B15 Symmetry properties of polytopes
Full Text:
##### References:
 [1] Merriell, D., Further remarks on concentric polygons, Amer. math. monthly, 72, 960-965, (1965) · Zbl 0132.14801 [2] Martini, H., On the theorem of napoleon and related topics, Math. semesterber., 43, 1, 47-64, (1996) · Zbl 0864.51009 [3] Vartziotis, D.; Wipper, J., On the construction of regular polygons and generalized napoleon vertices, Forum geom., 9, 213-223, (2009) · Zbl 1181.51020 [4] Petr, K., Ein satz über vielecke, Arch. math. phys., 13, 29-31, (1908) · JFM 39.0563.04 [5] Douglas, J., On linear polygon transformations, Bull. amer. math. soc., 46, 551-560, (1940) · JFM 66.0045.01 [6] Neumann, B., Some remarks on polygons, J. London math. soc., 16, 230-245, (1941) · JFM 67.1066.03 [7] Vartziotis, D.; Wipper, J., Classification of symmetry generating polygon-transformations and geometric prime algorithms, Math. pannon., 20, 2, 167-187, (2009) · Zbl 1240.51010 [8] Shephard, G., Sequences of smoothed polygons, (), 407-430 · Zbl 1051.52002 [9] Schuster, W., Regularisierung von polygonen, Math. semesterber., 45, 1, 77-94, (1998) · Zbl 0902.51014 [10] Ding, J.; Hitt, L.R.; Zhang, X.-M., Markov chains and dynamic geometry of polygons, Linear algebra appl., 367, 255-270, (2003) · Zbl 1024.51026 [11] Davis, P.J., Circulant matrices, (1994), Chelsea Publishing · Zbl 0898.15021 [12] Davis, P.J., Cyclic transformations of n-gons and related quadratic forms, Linear algebra appl., 25, 57-75, (1979) · Zbl 0408.15017 [13] Chang, G.; Davis, P.J., A circulant formulation of the napoleon – douglas – neumann theorem, Linear algebra appl., 54, 87-95, (1983) · Zbl 0529.51011 [14] Vartziotis, D.; Athanasiadis, T.; Goudas, I.; Wipper, J., Mesh smoothing using the geometric element transformation method, Comput. methods appl. mech. engrg., 197, 45-48, 3760-3767, (2008) · Zbl 1197.65197 [15] Vartziotis, D.; Wipper, J., The geometric element transformation method for mixed mesh smoothing, Engrg. comput., 25, 3, 287-301, (2009) [16] Vartziotis, D.; Wipper, J.; Schwald, B., The geometric element transformation method for tetrahedral mesh smoothing, Comput. methods appl. mech. engrg., 199, 1-4, 169-182, (2009) · Zbl 1231.74443
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.