Dynamic construction of a family of octic curves as geometric loci

Thierry Dana-Picard; Tomás Recio; Thierry Dana-Picard; Tomás Recio

doi:10.3934/math.2023993

AIMS Mathematics

2023, Volume 8, Issue 8: 19461-19476. doi: 10.3934/math.2023993

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Dynamic construction of a family of octic curves as geometric loci

Thierry Dana-Picard ^{1
,
,},
Tomás Recio ²

1.
Department of Mathematics, Jerusalem College of Technology, Havaad Haleumi Street 21, Jerusalem 9116011, Israel
2.
Universidad Antonio de Nebrija, C/Santa Cruz de Marcenado 27, 28015 Madrid, Spain

Received: 29 March 2023 Revised: 17 May 2023 Accepted: 18 May 2023 Published: 09 June 2023
MSC : 14H50, 51M15, 13P10

We explore the construction of curves of degree 8 (octics) appearing as geometric loci of points defined by moving points on an ellipse and its director circle. To achieve this goal we develop different computer algebra methods, dealing with trigonometric or with rational parametric representations, as well as through implicit polynomial equations, of the given curves. Finally, we highlight the involved mathematical or computational issues arising when reflecting on the outputs obtained in each case.
- plane curves,
- geometric locus,
- octic curve,
- interactive construction,
- dynamic geometry,
- GeoGebra,
- symbolic computation,
- Maple,
- implicitization
Citation: Thierry Dana-Picard, Tomás Recio. Dynamic construction of a family of octic curves as geometric loci[J]. AIMS Mathematics, 2023, 8(8): 19461-19476. doi: 10.3934/math.2023993

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Abstract

We explore the construction of curves of degree 8 (octics) appearing as geometric loci of points defined by moving points on an ellipse and its director circle. To achieve this goal we develop different computer algebra methods, dealing with trigonometric or with rational parametric representations, as well as through implicit polynomial equations, of the given curves. Finally, we highlight the involved mathematical or computational issues arising when reflecting on the outputs obtained in each case.

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