Moduli space of motions of geometric objects

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Computer Graphics produces a scene. It is a collection of large data, depending on the time variable.In this research, we study the structure of the space of scenes, in terms of geometry and Lie theory. The groups acting on the objects are the central role of this study. The example of such groups are Euclidean motion groups, orthogonal groups, rotation groups, and affine transformation groups. We give an interpretation of these groups into several subgroups or the matrix exponential image of subspaces as a basic tool of interpolation and blend in the graphics, such as SLERP (spherical linear interpolation) and ARAP (as rigid as possible). This analysis enables us to understand the previous method as well as to generalize and to improve the quality and speed of the interpolation.