In this paper the applications of equilibrium equation to geometric modeling is exploited and efficient numerical algorithms are proposed for solving the equilibrium equation. First we show that from diverse geometric modeling applications the equilibrium system can be extracted as the central framework. Second, by exploiting in-depth the special structures inherent in the geometric applications, we present simplified analytic solutions to the resulting geometric equilibrium equations via system decomposition. Finally, given the observation that the geometric equilibrium systems are extremely sensitive to both perturbations in input data and round off errors, efficient, stable and accurate numerical algorithms are proposed.