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Inflexions of Spiric Curves: A Tale of Two Tori

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Abstract

Cassini ovals (also called spiric curves or Spirics of Perseus) are an interesting family of bicircular quartic plane curves, appearing in various scientific fields, such as electric fields and delineation of influence zones of wells. They appear also as bisoptic curves of ellipses and, with some additional conditions, of hyperbolas. They provide connections between conic sections (intersection of a plane with a cone) and toric sections (intersection of a plane with a torus) with simultaneous manipulations on plane curves and surfaces in 3D. In general, Cassini ovals are presented as having one component, but the exploration of bisoptic curves of ellipses showed that despite the irreducibility of the polynomial, the curve is the union of two components, which cannot be distinguished by pure real algebraic methods. One of the components may have points of inflexion, and the other not. Networking between a computer algebra system and a dynamic geometry system, we explore the existence of these flexes, using the Hessian curve of the spiric, which is also a spiric. This exploration involves two tori. The work is also a new contribution about transition from 2D to 3D, and also to working in 2D and 3D together. The final section is devoted to educative remarks: on the one hand, the general issue of a multi-tools instrumental genesis and, on the other hand, the decision to study plane curves in a technology-rich environment. Collaboration between different kinds of software, including communication software has been reinforced by need of distance learning during the Covid-19 period.

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Notes

  1. Offsets appear also in architecture; see [21].

  2. Freely downloadable from http://www.geogebra.org.

  3. The polynomial equation describes both components, which are not distinguished by algebraic means, whence the name bisoptic for the curve.

  4. The reader can try this with a web applet. Recall that a construction with GeoGebra is based on free points and other objects, such as lines, circles, etc., may depend on these free points. Dragging (i.e. moving) a free point using the mouse induces changes in the dependent objects. These changes can be observed and analyzed; see [32] for a full study of the benefits of dragging points.

  5. For example Chijner’s applets such as https://www.geogebra.org/m/krmmnhu4#material/vttusatf.

  6. For example, http://mathcurve.com, and https://mathshistory.st-andrews.ac.uk/Curves/.

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Acknowledgements

This work has been partially supported by the CEMJ Chair at JCT.

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  1. Jerusalem College of Technology, Havaad Haleumi St. 21, 9116011, Jerusalem, Israel

    Thierry Dana-Picard

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Dana-Picard, T. Inflexions of Spiric Curves: A Tale of Two Tori. Math.Comput.Sci. 18, 2 (2024). https://doi.org/10.1007/s11786-024-00578-x

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