  We introduce a formalism to efficiently calculate lasing modes and optical
power flow in multi-section lasers with open boundaries. The formalism is
underpinned by a projection of the complex-valued electric field and its
spatial derivative onto a suitably extended complex $\mathcal{Z}$-plane, to
reduce the order of the problem and simplify analysis. In a single-section
laser, we show that a laser mode is a loxodrome on the extended complex
$\mathcal{Z}$-plane. In a multi-section laser, we obtain loxodromes for
individual sections of the laser. Then, a multi-section mode is constructed by
continuously concatenating individual loxodromes from each section using the
open boundary conditions. A natural visualization of this construction is given
by stereographic projection of the extended complex $\mathcal{Z}$-plane onto
the Riemann sphere. Our formalism simplifies analysis of lasing modes in open
multi-section lasers and provides new insight into the mode geometry and
degeneracy.
