Inextensible Surface Reconstruction Under Small Relative Deformations from Distributed Angle Measurements

A mathematical model to measure the shape of a 3D surface using angle measurements from embedded sensors is presented. The surface is known in a reference configuration and is assumed to have deformed inextensibly to its current shape. An inextensibility condition is enforced through a discretization of the metric tensor generating a finite number of constraints. This model allows to parameterize the shape of the surface using a small number of unknowns which leads to a small number of sensors. We study the singularities of the equations and derive necessary conditions for the problem to be well-posed as well as limitations of the algorithm. Simulations and experiments are performed on developable surfaces under relatively small deformations to analyze the performance of the method and to show the influence of the parameters used in our algorithm. Overall, the proposed method outperforms the current state-of-the-art by almost an order of magnitude.