Abstract: To solve the constraint equations of 7-section arc-type well trajectory design, mathematical elimination technique is used to eliminate the vector of angle holding section in well direction, and also the dog-leg angle, then simplify the conditions, and finally obtain the characteristic equation containing only one unknown (in most cases, it is a polynomial equation). All the real roots of the characteristic equation are obtained by solving the characteristic equation. It is mathematically proved that the the designed equations can be quickly solved by a set of analytical formulas containing the real roots of the characteristic equation. The algorithm can determine whether the design problem has a solution and how many solutions it has. In case of multiple solutions, these solutions can be obtained at the same time. The algorithm covers 45 meaningful combinations of unknowns about design problem in the 7-section system, and basically solves the numerical problem of the seven-section arc-type well trajectory design. Numerical examples prove that the algorithm is effective and can quickly get the solutions. The programming technique with polynomial classes can be used in the development of computer software for well trajectory design in drilling engineering.