Parametric Modeling of TECET Curves for Highway Geometric Design

Abstract

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 u?[0,1] 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.

Introduction

• Highway geometric design is essential for safety, comfort, and traffic efficiency.

• Proper horizontal and vertical alignment ensures smooth transitions and adequate sight distances, reducing accidents.

• In Mexico, these alignments follow the Highway Geometric Design Manual.

• A key element is the TECET curve—Tangent–Spiral–Circular–Spiral–Tangent—used to smoothly connect straight and curved road sections.

2. Purpose of the Study

• The article demonstrates the mathematical foundations behind TECET curves using concepts like calculus, vector analysis, and analytic geometry.

• A parametric model of the TECET curve is developed for implementation in visualization tools like GeoGebra, aiding academic learning and preliminary highway design.

3. TECET Curve Components

The TECET curve has three segments:

1. Entry Spiral (TE → EC): Curvature increases from 0 to 1/????_c.

2. Circular Arc (EC → CE): Constant curvature (radius = ????_c).

3. Exit Spiral (CE → ET): Curvature decreases from 1/????_c to 0.

? This design ensures C² continuity (position, direction, and curvature) for safe and smooth driving.

4. Key Parameters

• Degree of curvature (G_c): Determines the circular arc radius:
  ????_c = 1145.92 / G_c

• Spiral length (L_e) and arc length (L_c) define the full curve:

  • Total length: ????_T = 2L_e + L_c

  • Total deflection angle:
    ?_t = 2θ_e + ?_c
    where θ_e = L_e / (2R_c) and ?_c = L_c / R_c

• Clothoid Parameter (A):
  A = √(R_c · L_e)
  Controls how curvature increases linearly along the spiral:
  κ(s) = s / A²

5. Mathematical Formulation

• A dimensionless parameter (u ∈ [0, 1]) is used to normalize the entire curve length:

  • s(u) = u · L_T

  • Enables easy implementation in computational tools like GeoGebra.

???? Entry Spiral (Euler/Clothoid Spiral):

• Defined using Fresnel integrals:

  • x(u) = A√π · C(s(u)/A√π)

  • y(u) = A√π · S(s(u)/A√π)

• These provide smooth curvature progression from straight line to circular arc.

???? Circular Arc:

• Starts where entry spiral ends:

  • (X_E, Y_E) determined using spiral endpoint.

  • Arc center calculated via trigonometric relations.

• Parametrized using arc angle γ(u), allowing continuity and fixed curvature.

6. Significance

• The model captures smooth curvature transitions, minimizing abrupt lateral acceleration changes.

• By connecting math principles to practical road design, students and engineers can:

  • Understand the behavior of vehicles on curves,

  • Simulate trajectories,

  • Apply designs in tools like GeoGebra and CAD systems.

Conclusion

The compound Tangent–Spiral–Circular–Spiral–Tangent (TECET) curve represents a paradigmatic example of the connection between fundamental mathematics and applied civil engineering. The analysis developed in this work allows us to draw several conclusions: The use of the Euler spiral ensures that curvature varies linearly with arc length, providing a progressive increase or decrease in lateral acceleration. This results in safer and more comfortable trajectories for road users. Parametrization by means of a dimensionless parameter u?[0,1] facilitates the mathematical description of the TECET curve as a piecewise smooth function, enabling dynamic analysis of position, orientation, and curvature at any point along the trajectory. The presented formulation is compatible with symbolic and numerical computation tools such as GeoGebra, which makes it possible to visualize the behaviour of the curve as well as the velocity and acceleration vectors interactively. This opens didactic opportunities for early semesters of civil engineering education. The Frenet–Serret framework, applied to the parametrization of the curve, provides a solid basis for evaluating the kinematic components of a vehicle, showing how abstract notions of vector calculus have a direct application in the professional practice of highway design. Finally, the proposed methodology highlights the importance of mathematical training in civil engineering: courses such as calculus, analytic geometry, and vector analysis are not isolated subjects, but essential tools for understanding and developing solutions in real roadway infrastructure projects.

References

[1] Secretaría de Comunicaciones y Transportes (SCT). (2018). Manual de Proyecto Geométrico de Carreteras. Dirección General de Servicios Técnicos. https://www.sct.gob.mx/fileadmin/DireccionesGrales/DGST/Manuales/manual-pg/MPGC-2018-16-11-18.pdf [2] AASHTO. (2018). A Policy on Geometric Design of Highways and Streets (7th ed.). American Association of State Highway and Transportation Officials. [3] NIST. (2023). Fresnel Integrals. In NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/7.2 [4] do Carmo, M. P. (2016). Differential Geometry of Curves and Surfaces. Dover Publications. [5] Gray, A., Abbena, E., & Salamon, S. (2006). Modern Differential Geometry of Curves and Surfaces with Mathematica (3rd ed.). Chapman and Hall/CRC.

Copyright

Copyright © 2025 J. N. Zaragoza-Grife, A. C. Cabrera-Perez, R. C. Lopez-Sanchez, M. F. Chi-Cob. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.