To further study the mechanical property of a thin elastic rod, this paper will employ the Kichhoff equation which takes the form of Euler quaternion and study the topological configuration of the rod under compression. Adding the constraint condition to the differential equation, we can get a differential-algebraic equation(DAE).In order to be solved easily, DAE will be transformed into a c riterion form. To satisfy the boundary condition, we apply the shooting techniq ue to get the solution satisfying the boundary condition, and imitate the proced ure of pulling and pressing of thin elastic rod suffering a force. Simultaneousl y,to deal with the constraint stabilization phenomenon which is caused by errors , and according to the Euler quaternion character, we select a proper correction coefficient to keep the stability of the differential equation.