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In this paper, we show that the exact solution to time-optimal control of state-constrained second-order systems is uniquely determined by two curves: the forward velocity limitation curve and the backward velocity limitation curve. The forward (respectively, backward) velocity limitation curve stands for the curve beyond which, under the given control input and state constraints, the state cannot be steered forward in time from the initial point (respectively, backward in time from the final point). These two curves can be constructed by solving two scalar ordinary differential equations with no singularity at all and, therefore, their numerical construction can be done only with small computational burden by means of the well-known Euler or Runge-Kutta methods. This merit is demonstrated through its application to time-optimal control of robotic manipulators.