On the material and material-adapted approaches to curve framing with applications in path estimation, shape reconstruction, and computer graphics

Abstract

In this paper, we investigate an approach towards curve framing using material frames (MF). Motivated from the successful application of MF in shape sensing of rods in our previous work, we now present these frames as an alternative curve framing method. There are numerous instances of practical importance, where the dynamic system in consideration can be geometrically modeled by means of framed space curve. Unlike the Frenet-Serret and relatively parallel adapted frames (RPAF), the MF is conveniently defined in terms of the parameters associated with the system configuration.We detail the construction of the various material frames. We develop the relationships among the MF, Frenet frame, and the RPAF. We discuss the estimation of state space of the system from a limited set of material curvature and velocity data. In one of the approaches discussed, we obtain curvature-dependent shape functions to estimate the framed curve globally and discuss the errors associated with such estimations.We also describe the potential strengths of framed space curves in the reconstruction of slender structures, trajectory estimation of moving objects (like drone swarms), and in computer graphics. We do this by creating an analogy between the non-linear geometry of Cosserat beams and these applications.