Hermite Interpolation of Solid Orientations Based on a Smooth Blending of Two Great Circular Arcs on So(3)

An e cient algorithm is presented that generates a hermite interpolation quaternion curve. Given two unit quaternions q1; q2 2 SO(3), and two tangent vectors v1 and v2 at q1 and q2, respectively; two great circular arcs C1(t) and C2(t), 0 t 1, are constructed so that C1(0) = q1, C 0 1(0) = v1, C2(1) = q1, and C 0 2(1) = v2. The two great circular arcs C1 and C2 are then smoothly blended together on SO(3) to generate a hermite quaternion curve Q(t), 0 t 1, with the boundary conditions: Q(0) = q1; Q(1) = q2; Q0(0) = v1, and Q0(1) = v2. An e cient forward di erence method is developed to generate the great circular arcs. With the speed up thus obtained, the method of this paper and the squad method of Shoemake [16] now become the most e cient methods for the construction of quaternion curves which interpolate a given sequence of 3D solid orientations.