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Distance estimation from vision is fundamental for a myriad of robotic applications such as navigation, manipulation, and planning. Inspired by the mammal's visual system, which gazes at specific objects, we develop two novel constraints relating time-to-contact, acceleration, and distance that we call the $τ$-constraint and $Φ$-constraint. They allow an active (moving) camera to estimate depth efficiently and accurately while using only a small portion of the image. The constraints are applicable to range sensing, sensor fusion, and visual servoing. We successfully validate the proposed constraints with two experiments. The first applies both constraints in a trajectory estimation task with a monocular camera and an Inertial Measurement Unit (IMU). Our methods achieve 30-70% less average trajectory error while running 25$\times$ and 6.2$\times$ faster than the popular Visual-Inertial Odometry methods VINS-Mono and ROVIO respectively. The second experiment demonstrates that when the constraints are used for feedback with efference copies the resulting closed loop system's eigenvalues are invariant to scaling of the applied control signal. We believe these results indicate the $τ$ and $Φ$ constraint's potential as the basis of robust and efficient algorithms for a multitude of robotic applications.