Spatial camera orientation control by rotation-minimizing directed frames

The use of rotation-minimizing directed frames (RMDFs) for defining smoothly varying camera orientations along given spatial paths, in real or virtual environments, is proposed. A directed frame on a space curve r(xi) is a varying orthonormal basis (o, p, q) for R-3 such that o(xi) = r(xi)/vertical bar r(xi)vertical bar coincides with the unit polar vector from the origin to each curve point, and such a frame is rotation-minimizing if its angular velocity vector omega maintains a vanishing component along o. To facilitate computation of rotation-minimizing directed frames, it is shown that the basic theory is equivalent to the established theory for rotation-minimizing adapted frames-for which one frame vector coincides with the tangent t(xi) = r'(xi)/vertical bar r'(xi)vertical bar at each curve point-if one replaces the given space curve by its anti-hodograph (i.e., indefinite integral). A family of polynomial curves on which RMDFs can be computed exactly by a rational function integration, the Pythagorean (P) curves, is also introduced, together with algorithms for their construction. Copyright (C) 2009 John Wiley & Sons, Ltd.