Medical augmented reality has been actively studied for decades and many methods have been proposed to revolutionize clinical procedures. One example is the camera augmented mobile C-arm (CAMC), which provides a real-time video augmentation onto medical images by rigidly mounting and calibrating a camera to the imaging device. Since then, several CAMC variations have been suggested by calibrating 2D/3D cameras, trackers, and more recently a Microsoft HoloLens to the C-arm. Different calibration methods have been applied to establish the correspondence between the rigidly attached sensor and the imaging device. A crucial step for these methods is the acquisition of X-Ray images or 3D reconstruction volumes; therefore, requiring the emission of ionizing radiation. In this work, we analyze the mechanical motion of the device and propose an alternative method to calibrate sensors to the C-arm without emitting any radiation. Given a sensor is rigidly attached to the device, we introduce an extended pivot calibration concept to compute the fixed translation from the sensor to the C-arm rotation center. The fixed relationship between the sensor and rotation center can be formulated as a pivot calibration problem with the pivot point moving on a locus. Our method exploits the rigid C-arm motion describing a Torus surface to solve this calibration problem. We explain the geometry of the C-arm motion and its relation to the attached sensor, propose a calibration algorithm and show its robustness against noise, as well as trajectory and observed pose density by computer simulations. We discuss this geometric-based formulation and its potential extensions to different C-arm applications.