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

7 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
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

Bibliographical note

Funding Information:
The authors are sincerely grateful to the anonymous referees whose important suggestions allowed us to substantially improve the quality of this work. The authors want also to thank the anonymous referees for finding a shorter proof to Proposition 1 than the one initially suggested. Grant: This work was supported in part by the MEXT Global COE Program at Osaka University, Japan.

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

Access to Document

10.1016/j.cagd.2010.06.002

Cite this

@article{d9458af7c72141678d51c08506325209,

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",

note = "Funding Information: The authors are sincerely grateful to the anonymous referees whose important suggestions allowed us to substantially improve the quality of this work. The authors want also to thank the anonymous referees for finding a shorter proof to Proposition 1 than the one initially suggested. Grant: This work was supported in part by the MEXT Global COE Program at Osaka University, Japan.",

year = "2010",

month = oct,

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

language = "English",

volume = "27",

pages = "525--537",

journal = "Computer Aided Geometric Design",

issn = "0167-8396",

publisher = "Elsevier B.V.",

number = "7",

}

TY - JOUR

T1 - Complex Bézier curves and the geometry of polygons

AU - Ait-Haddou, Rachid

AU - Herzog, Walter

AU - Nomura, Taishin

N1 - Funding Information: The authors are sincerely grateful to the anonymous referees whose important suggestions allowed us to substantially improve the quality of this work. The authors want also to thank the anonymous referees for finding a shorter proof to Proposition 1 than the one initially suggested. Grant: This work was supported in part by the MEXT Global COE Program at Osaka University, Japan.

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 - https://www.scopus.com/pages/publications/77955467366

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

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Fingerprint

Cite this