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We present a new method by which to determine a three-dimensional space curve from its curvature and torsion. Conventionally, the task is done by solving the Frenet-Serret formulas to update the TNB frame and integrating the tangent vectors to update the node position. Our new method of curve reconstruction treats each curve segment as a segment of a helix. We use the given curvature and torsion of the curve segment to find the helix that the segment is supposed to take the form of. Then we update the TNB frame and node position using the basic properties of the helix. To validate our method, we generated an arbitrary space curve whose curvature and torsion are analytically given, and reconstructed it using both the old and new methods to compare their performances. In the simulation, using our new method appeared to offer more accurate results compared to the conventional 4th order Runge-Kutta method. This new method can be applied to fiber optic shape sensing for medical uses and many others.

This study proposes the use of strain sensors in a triple-helix configuration to measure the bending and twist deformation in a cylinder. Arranging the strain sensors on the cylinder in a triple-helix structure allows us to measure the twist and bending deformation simultaneously. The method consists of two steps: first to determine the local deformation factors from the surface strains, and second to reconstruct the overall deflected shape of the cylinder from the local deformation factors. We derived an exact analytical formula for the surface strain on a bent and twisted cylinder according to the deformation variables based on our original superhelix model, in which we regard the deformed cylinder segment as a helical coil and the sensors bound upon it as segments of superhelices. The local deflection rates are iteratively computed by the Newton-Raphson method using the surface strain formula. We incorporate the local deflection rates into the helical extension method, which is an exact solution to the Frenet-Serret formulas, to reconstruct the overall deflected shape of the cylinder. From the simulations, the proposed method was shown to determine the overall deformation state of a cylindrical body with remarkable precision. The position errors decreased rapidly with shorter spatial intervals of strain sensing and lower rates of the helical periodicity of the sensor arrays, showing a strong trend of convergence. Moreover, the pin-pointedness of the strain sensing was found to be one of the most crucial factors when attempting to ensure high accuracy of the shape estimation results. This study demonstrates a potential for general applicability to various fields, especially for the shape sensing of multicore optical fibers.

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