Abstract
We present two approaches to symbolically obtain isoptic curves in the dynamic geometry software GeoGebra in an automated, interactive process. Both methods are based on computing implicit locus equations, by using algebraization of the geometric setup and elimination of the intermediate variables. These methods can be considered as automatic discovery.
Our first approach uses pure computer algebra support of GeoGebra, utilizing symbolic differentiation of the input formula. Due to computational challenges we limit here our observations to quartic curves. The second approach hides all details in computer algebra from the user, that is, the input problem can be defined by a purely geometric way, considering a conic, a circle being given by its center and radius, and a parabola by the pair focus-directrix, for instance. The results are, however, not new, the novelty being is the way we obtain them, as a handy method for a new kind of man and machine communication. Both approaches deliver an algebraic output, namely, a polynomial and its graphical representation. The output is dynamically changed when using a slider bar. In this sense, dynamic study of isoptics can be introduced in a new way.
The internal GeoGebra computations, partly programmed by the authors, is an on-going work with various challenges in properly formulating systems of equations, in particular, to optimize computations and to avoid unnecessary extra curves in the output. Our paper highlights some of these challenges as well.
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Notes
- 1.
See the GeoGebra applet at https://www.geogebra.org/m/pwrWy9dG.
- 2.
See the GeoGebra applet at https://www.geogebra.org/m/SkQ5qxYr.
- 3.
See the GeoGebra applet at https://www.geogebra.org/m/tZcGGrCm.
- 4.
Valid according to the numerical precision of GeoGebra.
- 5.
For the same reason, with another CAS, the name bisoptic has been coined in [7].
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Acknowledgements
Second author partially supported by the grant MTM2017-88796-P from the Spanish MINECO (Ministerio de Economia y Competitividad) and the ERDF (European Regional Development Fund).
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Dana-Picard, T., Kovács, Z. (2018). Automated Determination of Isoptics with Dynamic Geometry. In: Rabe, F., Farmer, W., Passmore, G., Youssef, A. (eds) Intelligent Computer Mathematics. CICM 2018. Lecture Notes in Computer Science(), vol 11006. Springer, Cham. https://doi.org/10.1007/978-3-319-96812-4_6
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