Centro-affine dual curve pair of pre-tangent-dual and pre-normal-dual curves

This study examines a new transformation, called the equi-centro-transform, for curves defined within the frame of centro-affine differential geometry. By creating centro-affine dual curves of the pre-tangent-dual and pre-normal-dual curves under this transformation, the relationship between dual curve pairs is investigated geometrically and analytically. The primary objective of the study is to characterize the singularities that arise of a given curve and its centro-affine dual. Furthermore, the necessary and sufficient conditions for the formation of singularities in dual curve pairs are determined, and the behavior of singularities changing type under the transformation is analyzed in detail. Moreover, after a detailed examination of the pre-tangent dual curve and the pre-normal dual curve, these curves are compared with other related curve types to reveal their geometric relationships. At the end of the study, the theorems presented are supported with appropriate examples, the obtained theoretical results are concretized, and graphs of all examined curve types are drawn and presented visually using MATLAB.