Computing Quasi-Conformal Folds

Computing surface folding maps has numerous applications ranging from computer graphics to material design. In this work we propose a novel way of computing surface folding maps via solving a linear PDE. This framework is a generalization of the existing computational quasi-conformal geometry and allows precise control of the geometry of folding. This property comes from a crucial quantity that occurs as the coefficient of the equation, namely, the alternating Beltrami coefficient. This approach also enables us to solve an inverse problem of parametrizing the folded surface given only partial data with known folding topology. Various interesting applications such as fold sculpting on 3 dimensional models, study of Miura-ori patterns, and self-occlusion reasoning are demonstrated to show the effectiveness of our method.