Curves and their properties in dynamic geometry

In this thesis, the fundamental concepts of dynamic geometry within the framework of time scale theory are examined, and the adaptations of curve theory to time scales are discussed. The geometric applications of time scale theory, which unifies continuous and discrete analysis within a single framework, are investigated, and the analytical and geometrical properties of curves are evaluated. In the first part of the study, the concepts of analysis, differentiation, and integration on time scales are examined. The fundamental elements of time scale calculus and their roles in geometric applications are discussed. Subsequently, Frenet curves in dynamic geometry are investigated, and the properties of tangent, normal, and binormal vector fields associated with curves defined on time scales are analyzed. In the following chapters, plane and space curves in dynamic geometry are examined, and their parametric, explicit, and implicit representations are considered. Geometric interpretations of different representations of curves on time scales are presented, and the relationships among these representations are evaluated. Furthermore, tangent and normal lines of time scale curves are investigated, and their vector, parametric, explicit, and implicit equations are discussed. The study also examines the envelopes of plane curves and the concept of the osculating plane in order to investigate the local geometric behavior of curves. In addition, the curvature of time scale curves is considered, and its geometric significance is evaluated within the context of time scale theory. Finally, the Frenet formulas for time scale curves are examined, and their importance in curve geometry is highlighted.