Path connectivity of idempotents on a Hilbert space

Let $P$ and $Q$ be two idempotents on a Hilbert space. In 2005, J. Giol in [Segments of bounded linear idempotents on a Hilbert space, J. Funct. Anal. 229(2005) 405-423] had established that, if $P+Q-I$ is invertible, then $P$ and $Q$ are homotopic with $\tilde{s}(P,Q)\leq 2.$ In this paper, we have given a necessary and sufficient condition that $\tilde{s}(P,Q)\leq 2,$ where $\tilde{s}(P,Q)$ denotes the minimal number of segments required to connect not only from $P$ to $Q$, but also from $Q$ to $P$ in the set of idempotents.