  We show that a bilinear control system is approximately controllable if and
only if it is controllable in $\mathbb{R}^{n}\setminus\{0\}$. We approach this
problem by looking at the foliation made by the orbits of the system, and by
showing that there does not exist a codimension-one foliation in
$\mathbb{R}^{n}\setminus\{0\}$ with dense leaves that are everywhere
transversal to the radial direction. The proposed geometric approach allows to
extend the results to homogeneous systems that are angularly controllable.
