Complex Bézier curves and the geometry of polygons

Rachid Ait-Haddou; Walter Herzog; Taishin Nomura

doi:10.1016/j.cagd.2010.06.002

Complex Bézier curves and the geometry of polygons

Rachid Ait-Haddou^*, Walter Herzog, Taishin Nomura

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

In this paper, 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.

Original language	English (US)
Pages (from-to)	525-537
Number of pages	13
Journal	Computer Aided Geometric Design
Volume	27
Issue number	7
DOIs	https://doi.org/10.1016/j.cagd.2010.06.002
State	Published - Oct 2010
Externally published	Yes

Keywords

Bézier curve
Discrete Fourier transform
Geometry of polygons
Napoleon-Barlotti Theorem
Petr-Douglas-Neumann Theorem
Polar forms

ASJC Scopus subject areas

Modeling and Simulation
Automotive Engineering
Aerospace Engineering
Computer Graphics and Computer-Aided Design

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10.1016/j.cagd.2010.06.002

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title = "Complex B{\'e}zier curves and the geometry of polygons",

abstract = "In this paper, 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.",

keywords = "B{\'e}zier curve, Discrete Fourier transform, Geometry of polygons, Napoleon-Barlotti Theorem, Petr-Douglas-Neumann Theorem, Polar forms",

author = "Rachid Ait-Haddou and Walter Herzog and Taishin Nomura",

year = "2010",

month = oct,

doi = "10.1016/j.cagd.2010.06.002",

language = "English (US)",

volume = "27",

pages = "525--537",

journal = "Computer Aided Geometric Design",

issn = "0167-8396",

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TY - JOUR

T1 - Complex Bézier curves and the geometry of polygons

AU - Ait-Haddou, Rachid

AU - Herzog, Walter

AU - Nomura, Taishin

PY - 2010/10

Y1 - 2010/10

N2 - In this paper, 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.

AB - In this paper, 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.

KW - Bézier curve

KW - Discrete Fourier transform

KW - Geometry of polygons

KW - Napoleon-Barlotti Theorem

KW - Petr-Douglas-Neumann Theorem

KW - Polar forms

UR - http://www.scopus.com/inward/record.url?scp=77955467366&partnerID=8YFLogxK

U2 - 10.1016/j.cagd.2010.06.002

DO - 10.1016/j.cagd.2010.06.002

M3 - Article

AN - SCOPUS:77955467366

SN - 0167-8396

VL - 27

SP - 525

EP - 537

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

IS - 7

ER -

Complex Bézier curves and the geometry of polygons

Abstract

Keywords

ASJC Scopus subject areas

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