There are many systems such as beams, pipelines, coordinated drones swarm, DNA, etc., for which the configuration may be described by a framed space curve characterized by a single parameter. This research, therefore, utilizes the application of differential geometry and mechanics to investigate such systems. This work leads to the development of kinematically enhanced geometrically-exact beam theory, shape reconstruction of slender structures, path-estimation of a moving object, and computational geometry and graphics method.

The evolution of the system can be mathematically defined by a state space. An approach to approximate the state space of a single-manifold characteristic system using discrete material linear and angular velocity data is proposed. The methodology of path-estimation can be successfully applied to reconstruct the shape of deformed slender structures that captures the effect of curvature, shear, torsion, Poisson's deformation, warping, and axial deformation. The relationships are applied to generate some complicated structures like a double helix intertwined about a space curve, a leaf, and an entire plant.

Room for further improvisation of geometrically-exact beam theory was realized. A comprehensive kinematics of geometrically-exact beam subjected to a large deformation and finite strain is obtained. Among other deformation effects, the proposed kinematics also capture a fully coupled Poisson's and warping effects. The developed kinematics are ultimately used to establish a measurement model of discrete and finite length strain gauges attached to the beam.

The weak and strong form, Hamiltonian form, and Poisson bracket form of balance laws considering the enhanced kinematics of the beam are derived. The finite element model of the geometrically-exact beam with linear material properties is developed. Modal analysis is performed for a small deformation case.

The geometrically exact formulation discussed can be used to develop a reduced finite element model for DNA and bio-polymers. The shape sensing method has the potential to serve in the medical industry by helping in the location of surgical tubing, developing smart tethers that would help in the study of ocean surfaces, etc. Finally, the state-space estimation technique can be further extended to higher-order manifold problems like shape reconstruction of composite panels, membranes, distortion in space-time fabric, etc.