This paper describes how the concept of imposing geometric constraints by minimizing cost functions may be utilized and extended to accomplish a variety of animated modeling tasks for computer graphics. In this approach a complex 3-D geometric problem is mapped into a scalar minimization formulation. The mapping provides a straightforward method for converting abstract geometric concepts into a construct that is easily computed. The minimization approach is demonstrated in three application areas: computer animation, visualization, and physically-based modeling. In the computer animation application, cost minimization may be utilized to generate motion paths and joint parameters for animated actors. The approach may also be used to generate deformable models that extract closed 3-D geometric models from volume data for visualization. In the final application, the approach provides the fundamental structure to a physically-based model of woven cloth.