Abstract: This paper introduces the historical and practical significance of studying the brachistochrone problem and delves into the fastest motion of objects in a curved Earth tunnel. By establishing a mathematical model, the analytical expression of the brachistochrone curve is derived, and the physical meaning of the integral constant C is analyzed. The study reveals that curved tunnels, under certain conditions, can provide shorter transit times than straight tunnels, challenging intuitive perceptions. Further analysis yields the relationships among various physical quantities in the brachistochrone model, including the angular coordinate difference θab, the integral constant C, and the required transit time T. Additionally, with the assistance of Python visualization tools, this paper effectively presents graphical representations of the fastest paths in the model and illustrates the trends in the interrelationships among the physical quantities. This approach helps students better grasp the core concepts of the brachistochrone problem and the relationships between the various physical quantities involved in the model, fostering their ability to analyze and solve complex physical problems while building a clear physical picture and a comprehensive understanding．

Key words:  Earth tunnel, fastest motion, Python visualization toolkit