This letter tackles the problem of energy-efficient coverage path planning for exploring general surfaces by an autonomous vehicle. Efficient algorithms are developed to generate paths on freeform 3-D surfaces according to a special design pattern as height extremity aware Fermat spiral for this purpose. By using the exact boundary-sourced geodesic distances, the method for generating Fermat spiral paths is first introduced to cover a general surface. Then, heuristics for energy efficiency are incorporated to add peak points of a height field as sources for geodesic computation. The paths generated by our method can significantly reduce the cost caused by gravity. Physical experiments have been taken on different terrain surfaces to demonstrate the effectiveness of our approach.