Calibration of robot hand-eye generally needs to solve a rotation equation RaRx=RxRb. Many methods have been proposed

within which quaternion is the most concise one. But common methods using quaternion emphasize particularly on the application

and are short of relevant geometrical insight

and lack of comprehensive analysis of various solutions. In this paper we use quaternion geometry to solve the rotation equation

give proofs of solutions in various conditions in detail

and illuminate interesting insights between the analysis with quaternion matrix and the expression by geometry. Simulations have been tested. Analyzing solutions in various conditions and understanding the relevant geometrical meaning will help to ease the solving conditions and improve the performance of hand-eye calibration. Moreover

the study is important for the development of the quaternion geometrical analysis.