In the field of robotics, continuum manipulators are gaining attention due to their flexibility and ability to navigate complex environments. One of the key challenges in controlling these robotic systems is solving the null-space motion problem. This paper addresses this issue by proposing a bidirectional geometric iterative method based on the FABRIK (Forward and Backward Reaching Inverse Kinematics) algorithm to quickly solve the null-space motion of continuum manipulators. The abundance of degrees of freedom in continuum manipulators significantly complicates the null-space motion resolution. The FABRIK algorithm is renowned for its simplicity and efficiency in solving inverse kinematics problems for traditional robotic systems. By adapting this algorithm through bidirectional iteration, we tackle the unique challenges presented by continuum structures. Utilizing the geometric characteristics of continuum manipulators and iteratively adjusting joint configurations, our proposed method effectively computes feasible null-space motions, ensuring smooth and precise control of the manipulator.