We study the pion electromagnetic half-off-shell form factors $F_1$ and $F_2$ using an exactly solvable manifestly covariant model of a $(3+1)$ dimensional fermion field theory. The model provides a three-dimensional imaging of $F_1$ and $F_2$ as a function of $(Q^2,t)$, which are constrained by the Ward-Takahashi identity.

The normalization of the charge form factor $F_1$ is fixed by $F_1(Q^2=0, t=m^2_\pi)=1$ while the other form factor $F_2$ vanishes, i.e. $F_2(Q^2, t=m^2_\pi)=0$ for any value of $Q^2$ due to the time-reversal invariance of the strong interaction. The new form factor defined by $g(Q^2,t)=F_2(Q^2,t)/(t- m^2_\pi)$ is however measurable in the on-mass-shell limit. We note that $g(Q^2=0, t=m^2_\pi)$ is related with the pion charge radius.