We present AnalyticLC, a novel analytic method we developed for full light-curve modeling, and results we obtained by applying it to Kepler data. To our knowledge, this is the first analytic tool for transit variations that incorporates three-dimensional orbital motion. The analytic method also includes a previously unrecognized Transit Timing Variation (TTV) pattern, which arises in a near-resonant chain of three planets, a commonly observed architecture in the Kepler population. Our highly efficient and accurate implementation allowed full inversion of a large number of observed systems to derive physical and orbital parameters of 140 Kepler planets, including a description of their three-dimensional orbital geometry. We highlight our findings of dozens of Impact Parameter Variations (TbVs), which are fingerprints of mutual inclination in planetary systems. AnalyticLC can be further used for simultaneous fits of photometry and radial velocity measurements. It also offers advantages for analyzing observations with a long time span, a scenario becoming increasingly common in this era of multiple space missions. We have made AnalyticLC's code implementation publicly available.