Abstract
Within the area difference elasticity model, we numerically minimize the bending energy of phospholipid vesicles to study the nonaxisymmetric shapes at the oblate-prolate transition. We analyze the onset of a coherent structural hierarchy of hybrid shapes such as rackets, boomerangs, and starfish, which all consist of a flattened body and one or more elongated arms. The phase diagram of these shapes is characterized by a critical point terminating the line of discontinuous transitions between rackets and boomerangs/starfish. The critical point is located very close to the discocyte with the volume of a human erythrocyte, implying that in this part of the parameter space minute variations of volume and area difference can induce large changes of shape.