4 Rotations for N-Dimensional Graphics

Publisher Summary This chapter analyzes rotations for N -dimensional graphics. A family of techniques has been described for dealing with the geometry of N -dimensional models in the context of graphics applications. In this chapter, that framework is used to examine rotations in N -dimensional Euclidean space in greater detail. In particular, a natural N -dimensional extension is created both for the 3D rolling ball technique and for its analogous virtual sphere method. The defining property of any N -dimensional rolling ball (or tangent space) rotation algorithm is that it takes a unit vector v 0 ˆ = (0, 0,…., 0, 1) pointing purely in the N th direction (toward the “north pole” of the ball) and tips it in the direction of an orthogonal unit vector n ˆ = ( n 1 , n 2 ,…, n N −1 , 0) lying in the ( N −l)-plane tangent to the ball at the north pole. The chapter also addresses practical methods for specifying and understanding the parameters of N -dimensional rotations, and presents explicit 4D extensions of the 3D quaternion orientation splines.