zbMATH — the first resource for mathematics

Complex Bézier curves and the geometry of polygons. (English) Zbl 1217.68233
Summary: We associate to every planar polygon a complex polynomial, in which the blossom of the polynomial function captures the process in which linear transformations applied to the polygon lead to regular structures. In particular, we prove, in a purely algebraic way several well-known theorems on polygons such as the Napoleon-Barlotti Theorem, the Petr-Douglas-Neumann Theorem, and the Fundamental Decomposition Theorem of polygons to regular polygons.

68U07 Computer science aspects of computer-aided design
30C10 Polynomials and rational functions of one complex variable
65D17 Computer-aided design (modeling of curves and surfaces)
Full Text: DOI
[1] Barlotti, A., Una proprieta degli n-agoni che si ottengono transformando in una anita un n-agono regolare, Boll. un. mat. ital., 10, 96-98, (1955) · Zbl 0064.39806
[2] Connes, A., A new proof of Morley’s theorem, (), 43-46 · Zbl 1006.51010
[3] Douglas, J., Geometry of polygons in the complex plane, J. math. phys., 19, 93-130, (1940) · JFM 66.0728.03
[4] Fisher, J.C.; Ruoff, D.; Shiletto, J., Perpendicular polygons, Amer. math. monthly, 92, 1, 23-37, (1985) · Zbl 0562.51027
[5] Goldman, R.N., Dual polynomial bases, J. approx. theory, 79, 311-346, (1994) · Zbl 0824.41007
[6] Gray, S.B., Generalizing the petr-Douglas-Neumann theorem on n-gons, Amer. math. monthly, 110, 210-226, (2002) · Zbl 1053.51010
[7] Lester, J.A., Triangles I: shapes, Aequation math., 52, 30-54, (1996) · Zbl 0860.51009
[8] Marden, M., Geometry of polynomials, (1966), Amer. Math. Soc. Providence · Zbl 0173.02703
[9] Pech, P., The harmonic analysis of polygons and Napoleon’s theorem, J. geom. graph., 5, 1, 13-22, (2001) · Zbl 0991.51009
[10] Ramshaw, L., Blossoms are polar forms, Comp. aided geom. design, 6, 4, 323-358, (1989) · Zbl 0705.65008
[11] Schoenberg, I.J., The finite Fourier series and elementary geometry, Amer. math. monthly, 57, 390-404, (1950) · Zbl 0038.35602
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.