  The equilibrium shapes of vesicles are governed by the general shape equation
which is derived from the minimization of the Helfrich elastic free energy and
can be reduced to the Willmore equation in a special case. The general shape
equation is a high order nonlinear partial differential equation and it is very
difficult to find analytical solution even in axisymmetric case, which is
reduced to a seconder ordinary differential equation. Traditional axisymmetric
shape equation is with the turning radius as the variable. Here we study the
shape equation with the tangential angle as the variable. In this case, the
Willmore equation is reduced to the Bernoulli differential equation and the
general solution is obtained conveniently. We find that the curvature in this
solution is discontinuous in some cases, which was ignored by previous
researchers. This solution can satisfy the boundary conditions for open vesicle
with free edges.
