Rotations for N-Dimensional Graphics

In a previous article in Graphics Gems IV, \Geometry for N -Dimensional Graphics" (Hanson 1994), we described a fundamental family of techniques for dealing with the geometry of N -dimensional models in the context of graphics applications. Here, we build on that framework to look in more detail at the treatment of rotations in N dimensional Eucidean space. In particular, we give a previously overlooked but very natural N -dimensional extension of the 3D rolling ball technique described in an earlier Gem (Hanson 1992), along with the corresponding analog of the Virtual Sphere method (Chen et al. 1988). Next, we discuss a practical method for specifying and understanding the parameters of N -dimensional rotations in terms of nested hyperspheres. Finally, we give the 4D extension of 3D quaternion orientation splines along with some additional details of the 4D Arcball method (Shoemake 1994), and a discussion of the problems involved in extending these treatments to N -dimensional rotations with N > 4. For additional details and insights concerning N -dimensional geometry, we refer the reader to classic sources such as (Sommerville 1958,Coxeter 1991,Hocking and Young 1961,E mov and Rozendorn 1975).