Accurate navigation of flexible medical instruments like catheters require the knowledge of its pose, that is its position and orientation. In this paper multi-core fibers inscribed with fiber Bragg gratings (FBG) are utilized as sensors to measure the pose of a multi-segment catheter. A reconstruction technique that provides the pose of such a fiber is presented. First, the measurement from the Bragg gratings are converted to strain then the curvature is deduced based on those strain calculations. Next, the curvature and the Bishop frame equations are used to reconstruct the fiber. This technique is validated through experiments where the mean error in position and orientation is observed to be less than 4.69 mm and 6.48 degrees, respectively. The main contributions of the paper are the use of Bishop frames in the reconstruction and the experimental validation of the acquired pose.