FE models are based on mechanics which takes forces or displacements as input. In origami, the pattern is designed such that each portion of the mechanism comes with its own purpose. For example, the creases bend in a defined direction and the panels are meant to stay rigid. In other words, the geometry of different portions could be expected during the deformation. Since origami problems are defined by geometry, it is not necessary to convert the position input to the equivalent forces for stress -- strainstress–strain analysis. Instead, the FE formulation could be simplified to a geometry proximity and optimization, and directly consider the position. To allow the FE method to take positions as input and also output positions, a geometric formulation needs to be developed for origami. The formulation must be fast, stable and controllable, so that it can be applied to various designdesigns and used in an interactive environment. This section presents a geometry proximity function developed for the purpose, and a flowchart of the proposed framework is shown in Fig. 7.

When an external force is applied on a body, the dimension of the body is changed. The ratio of this dimensional change is the strain, which is a description of deformation excluding rigid-body motions. To model this physical phenomenon geometrically, it could measure the ``distance"distance between the differential of a deformation and the rotation group''group". In other words, a geometric model for minimizing the strain energy is to minimize the difference between the deformed and original shapes, and also compute the optimal orientation between them. For the whole body, its overall deformation is defined when the total strain energy of the system is minimized. This paper bases the development of the geometry proximity function on this principle.