Accurate distance estimation from monocular cameras is essential for intelligent monitoring systems. In many deployments, image coordinates are mapped to ground positions using planar homographies initialized by manual selection of corresponding regions. Small inaccuracies in this initialization propagate into systematic distance distortions. This paper derives an explicit relationship between homography perturbations and the resulting distance error, showing that the error grows approximately quadratically with the true distance from the camera. Based on this model, two simple correction strategies are evaluated: regression-based estimation of the quadratic error function and direct optimization of the homography via coordinate-based gradient descent. A large-scale simulation study with more than 19 million test samples demonstrates that regression achieves higher peak accuracy when the model is reliably fitted, whereas gradient descent provides greater robustness against poor initial calibration. This suggests that improving geometric calibration may yield greater performance gains than increasing model complexity in many practical systems.