The Brachistochrone is the curve that provides the fastest descent of a particle sliding without friction under a uniform gravitational field. Among all curves joining two fixed points, it minimizes the travel time and represents a fundamental problem in the calculus of variations and optimal control theory, with important applications in physics, engineering and optimization.

In this paper, we present explicit parametric representations of the Brachistochrone problem without initial velocity and perform a comparative analysis for several physical and mathematical configurations. Previous theoretical studies established the existence of optimal trajectories using dynamic programming methods, but without providing explicit expressions of the corresponding curves. Such representations are essential for both practical computations and theoretical investigations, since they allow accurate evaluation of physical quantities including distance, velocity, acceleration, curvature and travel time, both in the presence and absence of gravity.

The main objective of this work is to construct and compare parametric forms of the Brachistochrone curve without initial velocity and to analyze their geometric behavior, regularity properties, and physical interpretation. Particular attention is given to the influence of model parameters on the shape of the optimal trajectory and on the associated motion characteristics. The obtained results provide new insights into the structure of the problem and contribute to a better understanding of optimal trajectory design in gravitational environments.