Abstract: To improve slope-stability theory and calculation accuracy, this study constructs a slip-time functional using the calculus of variations and solves the Euler-Lagrange equation to derive an analytical solution for the critical slip surface of a homogeneous slope under self-weight. By introducing unit-weight and strength correction coefficients, a damping constant, and integral constants, a piecewise composite cycloid model is established to accommodate variations in stratigraphic parameters. Case comparisons between the circular-arc method and the proposed cycloid model are used to quantitatively validate model performance in terms of slip-surface fitting error, safety-factor deviation, and stress-zoning agreement. The results show that: (1) the critical slip surface of a gravity-driven slope is a brachistochrone rather than a circular arc; (2) for homogeneous slopes, the circular-arc model yields a slip-surface fitting error of 11.6% and a safety-factor error of 8.0%, whereas the cycloid model reduces these errors to 2.3% and 1.0%, respectively, and matches the three-stage slope failure pattern well; (3) for heterogeneous slopes, the circular-arc model has a fitting error of 18.2% and a safety-factor deviation of 15.5%, compared with 4.1% and 1.03% for the composite cycloid model, which accurately captures differentiated failure mechanisms; (4) the cycloid trajectory satisfies the variational extremum condition of the principle of least action and represents the optimal slip path. The cycloid model overcomes the mechanical deficiencies and low precision of traditional circular-arc methods and provides a new theoretical basis for slope-stability calculation, landslide mechanism analysis, and hazard-mitigation design.