  We investigate the observability of a general class of linear dispersive
equations on the torus $\mathbb{T}$. We take one line segment or two line
segments in space-time region as the observable set. We give the characteristic
on the slopes of the line segments to guarantee the qualitative observability
and quantitative observability respectively. The one line segment case, is
simple, follows directly from the Ingham's inequality. However, the two line
segments case is difficult, the statement of results and the proof rely heavily
on the language of graph theory. We also apply our results to (higher order)
Schr\"{o}dinger equations and the linear KdV equation.
