  We study the solution landscape and bifurcation diagrams of nematic liquid
crystals confined on a rectangle, using a reduced two-dimensional Landau--de
Gennes framework in terms of two geometry-dependent variables: half short edge
length $\lambda$ and aspect ratio $b$. First, we analytically prove that, for
any $b$ with a small enough $\lambda$ or for a large enough $b$ with a fixed
domain size, there is a unique stable solution that has two line defects on the
opposite short edges. Second, we numerically construct solution landscapes by
varying $\lambda$ and $b$, and report a novel X state, which emerges from
saddle-node bifurcation and serves as the parent state in such a solution
landscape. Various new classes are then found among these solution landscapes,
including the X class, the S class, and the L class. By tracking the Morse
indices of individual solutions, we present bifurcation diagrams for nematic
equilibria, thus illustrating the emergence mechanism of critical points and
several effects of geometrical anisotropy on confined defect patterns.
