  In 3D nematic liquid crystals, disclination lines have a range of geometric
structures. Locally, they may resemble $+1/2$ or $-1/2$ defects in 2D nematic
phases, or they may have 3D twist. Here, we analyze the structure in terms of
the director deformation modes around the disclination, as well as the nematic
order tensor inside the disclination core. Based on this analysis, we construct
a vector to represent the orientation of the disclination, as well as tensors
to represent higher-order structure. We apply this method to simulations of a
3D disclination arch, and determine how the structure changes along the contour
length. We then use this geometric analysis to investigate three types of
forces acting on a disclination: Peach-Koehler forces due to external stress,
interaction forces between disclination lines, and active forces. These results
apply to the motion of disclination lines in both conventional and active
liquid crystals.
