  A two-dimensional or quasi-two-dimensional nematic liquid crystal refers to a
surface confined system. When such a system is further confined by external
line boundaries or excluded from internal line boundaries, the nematic
directors form a deformed texture that may display defect points or defect
lines, for which winding numbers can be clearly defined. Here, a particular
attention is paid to the case when the liquid crystal molecules prefer to form
a boundary nematic texture in parallel to the wall surface (i.e., following the
homogeneous boundary condition). A general theory, based on geometric argument,
is presented for the relationship between the sum of all winding numbers in the
system (the total winding number) and the type of confinement angles and curved
segments. The conclusion is validated by comparing the theoretical defect rule
with existing nematic textures observed experimentally and theoretically in
recent years.
