Development of Frenet-Serret Frame and the Apollonian Window

Abstract The present study focuses on Frenet-Serret frame and the Apollonian Window. In the first part of the In the first part of the study Apollonian disks are generated for first four generations by developing visual basic codes in excel. For the second part of the study, three orthonormal basis vectors, namely, tangent, normal, and binormal vectors have been calculated for the tangent points of Apollonian discs for the first three generations. Equations of the Normal, Osculating and Rectifying planes and Taylor Series approximation have been calculated for specific θs. Because Apollonian Window consists of planar curves with constant curvature, torsion is nowhere present. The planar Frenet-Serret Equations for the first three generations for the Apollonian Window is also shown.