1 4. 3D Hierarchies for Animation Ronan Boulic and Olivier Renault Computer Graphics Laboratory Swiss Federal Institute of Technology Lausanne, Switzerland Geometric representation associated with computer-assisted modeling has proven to be very usefull to visualize and understand the structure and behavior of a wide spectrum of entities (Foley et al. 1990). But if we examine the scope of the current 3D geometric models, we notice that they tend to embrace a static view of the information structure. In this chapter, we are particularly interested in stating the requirements for the modeling of complex 3D environments with mobile components. Such a modeling approach has to integrate a dynamic view of these components such that specific goals can be acheived in space and time. A key requirement is to provide a representation of relative motions and this can be addressed through the use of hierarchical modeling. In the first section we review some geometric models and the PHIGS graphic standard which could be considered suitable for the modeling of 3D hierarchies. Then, we focus on the representation of mobility as it was first studied in robotics. The following sections are dedicated to an overview of the related technical requirements and to their application to a data hierarchy. The last section covers the general techniques of interaction and animation of 3D hierarchies . 4.1. Existing Hierarchical Geometric Models Some geometric models are, by definition, 3D hierarchies. This is the case of the octree and CSG representations. We now present briefly these hierarchical approaches and then study their scope and their adequacy to the requirement of relative motion representation. 4.1.1. Octree The octree representation is a subset of the cell decomposition technique, in which a solid is decomposed into arbitrary cells and is represented by each cell in the decomposition. Octree representation gives a 3D systemization to this cell decomposition.