This paper presents a geometric transversals approach for representing the probability of track detection as an analytic function of time and target motion parameters. By this approach, the optimization of the detection probability subject to sensor kinodynamic constraints can be formulated as an optimal control problem. Using the proposed detection probability function, the necessary conditions for optimality can be derived using calculus of variations, and solved numerically using a variational iteration method (VIM). The simulation results show that sensor state and control trajectories obtained by this approach bring about a significant increase in detection probability compared to existing strategies, and require a computation that is significantly reduced compared to direct methods.