Construction methods are presented that generate Hermite interpolation quaternion curves on SO(3). Two circular curves C1(t) and C2(t), 0 ≤ t ≤ 1, are generated that interpolate two orientations q1 and q2, and have boundary angular velocities: C1′(0) = ω1 and C2′(1) = ω2, respectively. They are smoothly blended together on SO(3) to generate a Hermite quaternion curve Q(t) ∈ SO(3), 0 ≤ t ≤ 1, which satisfies the boundary conditions: Q(0) = q1, Q(1) = 2, Q′(0) = ω1, and Q′ (1) = ω2.