  By studying the group of rigid motions, $PSH(1)$, in the 3D-Heisenberg group
$H_1$, we define the density and the measure for the sets of horizontal lines.
We show that the volume of a convex domain $D\subset H_1$ is equal to the
integral of length of chord over all horizontal lines intersecting $D$. As the
classical result in integral geometry, we also define the kinematic density for
$PSH(1)$ and show the probability of randomly throwing a vector $v$ interesting
the convex domain $D\subset D_0$ under the condition that $v$ is contained in
$D_0$. Both results show the relationship connecting the geometric probability
and the natural geometric quantity in Cheng-Hwang-Malchiodi-Yang's work
approached by the variational method.

Differential Geometry