This paper presents a new method to produce computationally efficient models of robots that have planar elastic flexure joints. An accurate, low-dimensional model of large deformation bending is important to precisely describe the configuration of a flexure-jointed manipulator. The new model is based on the assumption that the curvature of a beam in bending is smooth and, thus, can be approximated by low-order polynomials. This produces a description of flexure motion that can be used as a joint model when expressed as a homogeneous transformation between rigid links--essentially a “drop in” replacement for traditional joint models such as screw coordinates and Denavit-Hartenberg conventions. Derivatives of the joint kinematics such as Jacobians and Hessians are accurate and easy to compute. We will show that with only three parameters, this model faithfully reproduces the elastic deformation of a flexure hinge predicted by the continuum model, even for large angles, without requiring numerical integration or many finite elements. The model can also be used to accurately compute the compliance and compressive buckling load of the flexure, as predicted by the continuum model.