Abstract
In geometry, knowing the differential properties of a curve at a point is very important, as it would help us understand its behavior around that point; even more so when said curve is generated by the intersection of hypersurfaces in 
. This paper describes a new Mathematica package, Frenet4D, that allows to visualize and calculate the Frenet frame of the curve given by the transversal intersection of three implicit hypersurfaces in 
thus like the visualization of the respective hypersurfaces. The output obtained is consistent with Mathematica’s notation and results. To show the performance of the package, several illustrative and interesting examples are described.
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References
Anto, L.A., Fiestas, A.M., Ojeda, E.J., Velezmoro, R., Ipanaqué, R.: A mathematica package for plotting implicitly defined hypersurfaces in \(\mathbb{I\!R}^4\). In: Gervasi, O., et al. (eds.) ICCSA 2020. LNCS, vol. 12250, pp. 117–129. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58802-1_9
Badr, S.A.-N., Abdel-All, N.H., Aléssio, O., Düldül, M., Düldül, B.: Non-transversal intersection curves of hypersurfaces in Euclidean 4-space. J. Comput. Appl. Math. 288, 81–98 (2015)
Burgos, G., Jiménez, J., Ascate, Y.: Frenet Serret apparatus calculation of curves given by the intersection of two implicit surfaces in \(\mathbb{I\!R}^3\) using Wolfram Mathematica v. 11.2. Sel. Matemáticas 6(2), 338–347 (2019)
Jiménez, J.: Calculation of the differential geometry properties of implicit parametric surfaces intersection. Revista Científica Pakamuros 8(1), 69–79 (2020)
Jiménez-Vilcherrez, J.K.: Calculation of the differential geometry properties of implicit parametric surfaces intersection. In: Gervasi, O., Murgante, B., Misra, S., Garau, C., Blečić, I., Taniar, D., Apduhan, B.O., Rocha, A.M.A.C., Tarantino, E., Torre, C.M., Karaca, Y. (eds.) ICCSA 2020. LNCS, vol. 12251, pp. 383–394. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58808-3_28
Aléssio, O., Düldül, M., Düldül, B.U., Badr, S.A.-N., Abdel-All, N.H.: Differential geometry of non-transversal intersection curves of three parametric hypersurfaces in Euclidean 4-space. Comput. Aided Geom. Des. 31(9), 712–727 (2014)
Aléssio, O.: Differential geometry of intersection curves in R4 of three implicit surface. Comput. Aided Geom. Des. 26(4), 455–471 (2009)
Patrikalakis, N.M., Maekawa, T.: Shape Interrogation for Computer Aided Design and Manufacturing, 1st edn. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04074-0
Seok, H., Tae-wan, K., Cesare, B.: Comprehensive study of intersection curves in R4 based on the system of ODEs. J. Comput. Appl. Math. 256, 121–130 (2014)
Torrence, B.F., Torrence, E.A.: The Student’s Introduction to Mathematica and the Wolfram Language, 3rd edn. Cambridge University Press, Cambridge (2019)
Uyar Düldül, B., Düldül, M.: The extension of Willmore’s method into 4-space. Math. Commun. 17(2), 423–431 (2012)
Velezmoro, R., Ipanaqué, R., Mechato, J.A.: A mathematica package for visualizing objects Inmersed in \(\mathbb{I\!R}^{4}\). In: Misra, S., et al. (eds.) ICCSA 2019. LNCS, vol. 11624, pp. 479–493. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-24311-1_35
Williams, M.Z., Stein, F.M.: A triple product of vectors in four-space. Math. Mag. 37(4), 230–235 (1964)
Xiuzi, Y., Takashi, M.: Differential geometry of intersection curves of two surfaces. Comput. Aided Geom. Des. 16(8), 767–788 (1999)
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Jiménez-Vilcherrez, J.K., Velezmoro-León, R., Ipanaqué-Chero, R. (2021). Calculation of the Differential Geometry of the Intersection of Implicit Hypersurfaces in 
with Mathematica.
In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2021. ICCSA 2021. Lecture Notes in Computer Science(), vol 12953. Springer, Cham. https://doi.org/10.1007/978-3-030-86976-2_4
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DOI: https://doi.org/10.1007/978-3-030-86976-2_4
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