We calculate the magnetic field generated by a steady current that takes the shape of two types of special curves: hypocycloids and epicycloids with n numbers of sides. The computation was performed in the center of the referred curves. For this purpose, we use the Biot-Savart law which is studied in every introductory-level electricity and magnetism course. The result is quite general because it is obtained as a function of the number of sides of the curve and in terms of a parameter є that identifies the type of curve considered (є = —1 hypocycloids and є = +1 epicycloids).
Abad, D. R. (2021). The Magnetic field for an n-cusped Epi-and Hypo-Cycloids loop current. Revista Brasileira de Ensino de Fisica, 43, 1–5. https://doi.org/10.1590/1806-9126-RBEF-2020-0482