  The parity operator $\cal P$ and time reversal operator $\cal T$ are two
important operators in the quantum theory, in particular, in the $\cal
PT$-symmetric quantum theory. By using the concrete forms of $\cal P$ and $\cal
T$, we discuss their geometrical properties in two dimensional spaces. It is
showed that if $\cal T$ is given, then all $\cal P$ links with the quadric
surfaces; if $\cal P$ is given, then all $\cal T$ links with the quadric
curves. Moreover, we give out the generalized unbroken $\cal PT$-symmetric
condition of an operator. The unbroken $\cal PT$-symmetry of a Hermitian
operator is also showed in this way.
