Abstract In a lot of applications, the complete orientation in 3-d Euclidean space is not necessary to fulfill a given task. In these tasks, only one direction of the target orientation is important. These problems are called Incomplete Orientation Constraint Problems. Aiming for such problems, we define the foot of perpendicular and the perpendicular curve of geodesic in rotation group. We also present the analytical s for the foot of perpendicular and the perpendicular curve. Based on this theory, we propose a general point-to-point trajectory generator algorithm by following the perpendicular curve. Our algorithm can not only be applied to manipulators with 6 DOFs, but also be applied to manipulators with 5-DOFs or 4-DOFs. The experiments in simulation are conducted on a PUMA560 with 6-DOFs. To verify the generality of our method, we fix the 6-th and 4-th joints successively to get 5-DOF and 4-DOF manipulators. The experimental results verify the advantage of our method in handling incomplete orientation constraint problem.