The inherent compliance of continuum robots holds great promise in the fields of soft manipulation and safe human-robot interaction. This compliance reduces the risk of damage to the manipulated object and the surroundings. However, continuum robots have theoretically infinite degrees of freedom, and this high flexibility usually leads to complex deformations with external forces and positional constraints. How to describe this complex deformation is the main challenge for modelling continuum robots. In this study, we investigated a novel variable curvature modeling method for continuum robots, considering external forces and positional constraints. The robot configuration curve is described by the developed mechanics model, and then the robot is fitted to the curve. To validate the model, a 10-section continuum robot prototype with a length of 1 m was developed. The ability of the robot to reach the target points and track complex trajectories with load verified the feasibility and accuracy of the model. The ratio of the average position error of the robot endpoint to the robot length was less than 2.38%. This work may serve a new perspective for design analysis and motion control of continuum robots.