Abstract: The design of three-dimensional borehole trajectories using the natural curve approach under tangent-to-point conditions can be attributed to solving a system of multivariate highly nonlinear equations. Conventional numerical iterative methods often face challenges such as the inability to determine suitable initial iteration values. To overcome this issue in a stable and efficient manner, an univariate nonlinear equation (characteristic equation) was derived from this system. Once all real roots of the characteristic equation are determined, when the well deviation angle is prioritized, all the other variables can be calculated using a set of analytical formulas. In cases where the azimuth angle is prioritized, one of the other variables requires solving a simple trigonometric equation, while the remaining variables can be calculated using a set of analytical formulas. The characteristic function is a multimodal continuous function with numerous real roots. To enhance computational efficiency, the constraints were employed to define the maximum permissible interval for meaningful real roots, and the characteristic equation was solved within this interval using root separation and the bisection method. Numerical cases demonstrate that the proposed algorithm effectively and quickly solves the borehole trajectory design problem, eliminating the need for manually assigned initial iteration values. Moreover, it effectively addresses the issue of multiple solutions in a set of constraint equations. The results are fully consistent with those produced by Compass, a commercial drilling design software. This method is quite practical and can be integrated into the development of domestic and alternative drilling design software.