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Soft robot are celebrated for their propensity to enable compliant and complex robot-environment interactions. Soft robotic manipulators, or slender continuum structure robots have the potential to exploit these interactions to enable new exploration and manipulation capabilities and safe human-robot interactions. However, the interactions, or perturbations by external forces cause the soft structure to deform in an infinite degree of freedom (DOF) space. To control such system, reduced order models are needed; typically models consider piecewise sections of constant curvature although external forces often deform the structure out of the constant curvature hypothesis. In this work we perform an analysis of the trade-off between computational treatability and modelling accuracy. We then propose a new kinematic model, the Piecewise Affine Curvature (PAC) which we validate theoretically and experimentally showing that this higher-order model better captures the configuration of a soft continuum body robot when perturbed by the external forces. In comparison to the current state of the art Piecewise Constant Curvature (PCC) model we demonstrate up to 30\% reduction in error for the end position of a soft continuum body robot.