We present a novel framework for acquisition and reconstruction of 3D curves using orientations provided by inertial sensors. While the idea of sensor shape reconstruction is not new, we present the first method for creating well-connected networks with cell complex topology using only orientation and distance measurements and a set of user- defined constraints. By working directly with orientations, our method robustly resolves problems arising from data inconsistency and sensor noise. Although originally designed for reconstruction of physical shapes, the framework can be used for “sketching” new shapes directly in 3D space. We test the performance of the method using two types of acquisition devices: a standard smartphone, and a custom-made device.