  In a reaction to excite the resonant state followed by the sequential
cluster-decay, the in-plane angular correlation method is usually applied to
determine the spin of the mother nucleus. However, the correlation pattern
exhibited in a two-dimensional angular-correlation spectrum depends on the
selected coordinate system. Particularly the parity-symmetric and the
axial-symmetric processes should be presented in a way to enhance the
correlation pattern whereas the non-symmetric process should be plotted
elsewhere in order to reduce the correlation background. In this article, three
possible coordinate systems, which were previously adopted in the literature,
are described and compared to each other. The consistency of these systems is
evaluated based on the real experimental data analysis for the 10.29-MeV state
in $^{18}$O. A spin-parity of 4$^+$ is obtained for all three coordinate
systems.
