The Smartest Way to Orient a Solar Panel

A mathematical framework for optimally orienting solar panels to maximize energy harvest under any lighting condition is presented. Starting from Lambert's cosine law and irradiance fundamentals, the problem is reframed as a spherical convolution between the radiance function and a cosine kernel. Rather than using an expensive fisheye camera, the approach uses photo detectors tilted at a sufficiently large angle relative to the panel. Fourier analysis reveals that the photo differential signal is equivalent to the gradient of a blurred irradiance function, which eliminates local maxima and enables gradient ascent to reliably find the global optimum. A prototype system tested in five scenarios — direct sunlight, cloudy conditions, urban shading, building reflections, and indoor environments — outperforms both sun-tracking and fixed-orientation baselines, harvesting up to 10% more energy in shaded urban settings.