Helical extension method for solving the natural equation of a space curve

Abstract

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.