Parametric Modeling of TECET Curves for Highway Geometric Design

This manuscript presents an academic discussion on the use of mathematical tools for the parametric modeling of horizontal road alignments, specifically the Tangent–Spiral–Circular–Spiral–Tangent (TECET) curve. The objective is to show how fundamental concepts of calculus and differential geometry—arc length, curvature, Frenet–Serret frame, and Fresnel integrals—form the analytical basis for geometric design criteria established in road standards. A piecewise smooth parametric model is constructed, using a dimensionless parameter ࢛૙] ∋ , [૚ that maps the total curve length, enabling the evaluation of vehicle kinematics at any point of the alignment. The model ensures continuity of position, tangent direction, and curvature (C2 continuity) at the junctions between spirals and circular arc. An example with data from the SCT (Mexico) is included, implemented in GeoGebra to allow dynamic visualization of tangent and acceleration vectors. This work seeks to highlight the link between mathematics taught in the early semesters of civil engineering and its concrete application in advanced road design projects.