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A Brief Note on the Approach to the Conic Sections of a Right Circular Cone from Dynamic Geometry

Abstract

Nowadays there are different powerful 3D dynamic geometry systems (DGS) such as GeoGebra 5, Calques 3D and Cabri geometry 3D. An obvious application of this software that has been addressed by several authors is obtaining the conic sections of a right circular cone: the dynamic capabilities of 3D DGS allows to slowly vary the angle of the plane w.r.t. the axis of the cone, thus obtaining the different types of conics. In all the approaches we have found, a cone is firstly constructed and it is cut through variable planes. We propose to perform the construction the other way round: the plane is fixed (in fact it is a very convenient plane: \(z=0\)) and the cone is the moving object. This way the conic is expressed as a function of x and y (instead of as a function of x, y and z). Moreover, if the 3D DGS has algebraic capabilities, it is possible to obtain the implicit equation of the conic.

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Acknowledgements

This work was partially supported by the research Projects TIN2015-66471-P (Government of Spain) and CASI-CAM S2013/ICE-2845 (Comunidad Autónoma de Madrid) and the Research Group ACEIA. We would also like to thank the anonymous referees for their most valuable comments and suggestions, that helped to improve and clarify the paper.

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Correspondence to Eugenio Roanes-Lozano.

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Roanes-Lozano, E. A Brief Note on the Approach to the Conic Sections of a Right Circular Cone from Dynamic Geometry. Math.Comput.Sci. 11, 439–448 (2017). https://doi.org/10.1007/s11786-017-0307-3

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Keywords

  • Conics
  • Dynamic geometry
  • 3D geometry
  • Computer algebra

Mathematics Subject Classification

  • Primary 68U99
  • Secondary 97G40
  • Tertiary 68U05