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Exploring the Isoptics of Fermat Curves in the Affine Plane Using DGS and CAS

  • Thierry Dana-PicardEmail author
  • Aharon Naiman
  • Witold Mozgawa
  • Waldemar Cieślak
Article
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Abstract

Isoptic curves of plane curves are a live domain of study, mostly for closed, smooth, strictly convex curves. A technology-rich environment allows for a two-fold development: dynamical geometry systems enable us to perform experiments and to derive conjectures, and computer algebra systems (CAS) are the appropriate environments for an algebraic approach for determining isoptics, with its Gröbner bases packages, amongst others. Closed Fermat curves in the affine plane present a specific problem: the variables of the involved polynomials represent the coordinates of points on the Fermat curve, which avoid the dense set of points with two rational coordinates (excepted 4 “trivial” points). Therefore, automated computation has to be performed over the field \(\mathbb {R}\) of the real numbers, with which the Gröbner bases packages do not work. Instead, other packages implemented in CASs have to be used. First, we present a study of the orthoptics of closed Fermat curves of even order. Then we proceed to an algebraic study of these curves using a CAS. The generalization to angles other than \(90^\circ \) is performed afterwards, with an algebraic approach using support functions, then using numerical methods.

Keywords

Isoptic curves Fermat curves Automated methods Conputer algebra system Dynamic geometry system 

Mathematics Subject Classification

53A04 53A15 51M15 65–04 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Thierry Dana-Picard
    • 1
    Email author
  • Aharon Naiman
    • 1
  • Witold Mozgawa
    • 2
  • Waldemar Cieślak
    • 3
  1. 1.Jerusalem College of TechnologyJerusalemIsrael
  2. 2.Maria Curie-Skłodowska UniversityLublinPoland
  3. 3.Lublin University of TechnologyLublinPoland

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