Abstract: Assuming that there exist the bodies A and B in Keplerian orbits around a single gravitational center and a spacecraft transfers from A to B, a new model called v_∞-transfer-orbit (VTO)-problem is proposed for determining the spacecraft’s transfer orbit. In the VTO-problem, the escaping time t₀ and the escaping velocity v_∞ departing from A are selected as the spacecraft’s orbital determination parameters. According to the spatial relative positions between A and B, the VTO-problem is divided into three cases: A/B is nonplanar, A/B is coplanar, and A/B is co-orbital, and there exist three types of solutions: General-VTO, Backflip-VTO and Resonant-VTO. In this paper, a uniform geometric analysis method for solving the VTO-problem is introduced, in which the position constraint of the spacecraft’s arrival at B is decomposed into orbital constraint and time constraint, the spacecraft’s orbital parameters are resolved by a single variable based on the orbital constraint, and an equation referring to this single variable is constructed based on the time constraint. According to the geometric analysis method, the VTO-problem is transformed into a one-variable equation-rooting problem. Firstly, the one-variable equation for General-VTO is derived in response to the cases of A/B nonplanar, A/B coplanar, and A/B co-orbital, and the intervals of the variable and an efficient equation-rooting algorithms based on the cubic spline interpolation are elaborated. Secondly, the different one-variable equation-rooting problem for Backflip-VTO is derived, and another set of equation-rooting algorithms are described on the basis of analyzing the equation function properties, such as monotonicity, extreme points and inflection points. Thirdly, the analytic solution is given directly for Resonant-VTO. Finally, examples are given to expound the solution multiplicity of the VTO-problem.