The Magnetic field for an n-cusped Epi-and Hypo-Cycloids loop current

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 ϵ that identifies the type of curve considered (ϵ = −1 hypocycloids and ϵ = + 1 epicycloids).