A Unified Tangent-Based Parametrization of Conic Sections Generated from a Reference Circle

This study presents a unified geometric approach for determining the parameters of conic sections based on tangent geometry. Four generalized equations are derived that de scribe circles, ellipses, parabolas, and hyperbolas through the angular relationships between tangent lines, their points of contact, and the associated polar angle with respect to a central reference circle. The proposed framework establishes a direct geometric link between the tangent an gles and the emergence of different conic types, extending the classical polar-angle-based description of conic sections. The derived equations admit canonical conic forms, allowing the geometric parameters and classification of each conic to be identified within a single parametric structure. Moreover, the formulation reveals that all four conic sections intersect at the external point from which the tangents to the reference circle are drawn, providing a coherent ge ometric interpretation of their common origin. This tangent-based parametrization offers a mathematically consistent framework for analytical geometry and presents potential rele vance for applications in computational geometry and structural modeling.