Abstract: For the important application of C³ continuity in engineering science,the C³ spline interpolation is constructed by PH curves of degree seven.Based on the particular expression of PH curve of degree seven and boundary condition of spline interpolation,the problem of constructing spline interpolation is changed to finding the solution of quadratic complex equations related with complex variables;Because the solutions of quadratic complex equations are not unique,in order to avoid the problem of jumping from tracking path to another in traditional homotopy method,adaptive homotopy method is proposed to obtain all the solutions of quadratic complex equations.The experimental results show that this method can effectively obtain all the spline interpolations,which satisfy the given conditions.