Based on the Helfrich elastic curvature energy model, the stable shapes for the two patterns of three-domain phase separation are studied in detail for the experimental parameters with direct minimization method in order to explain the interesting experimental results by Yanagisawa et al. (2010 Phys. Rev. E 82 051928). According to their experimental results, there are two transition processes. In the first process, the three-domain vesicles are formed, which are metastable. After several tens of minutes, the three-domain vesicles begin to bud, which is the second process. In the first process, the three-domain vesicles are formed with two patterns. The pattern with the liquid-ordered (Lo) phase in the middle with roughly cylindrical shape and two cap-shape liquid disordered (Ld) domains on each side of the Lo domain is termed pattern I in our paper, and the pattern with Ld domain in the middle with roughly cylindrical shape and two cap-shape Lo domains on each side is referred to as pattern Ⅱ. In the same paper of M. Yanagisawa et al., an approximate calculation is made with the vesicle shapes of the two patterns approximately represented by spheroids. Their calculation shows that the transition point of the two patterns is at o* 0.27 in the case of = 0.02 (or v = 0.942) and = 50, in contrast with the experimental result of o* 0.5. Here o is the area fraction of Lo phase, and is the excess area (which is usually represented by reduced volume v in the previous literatures), is the reduced line tension at the boundary of two adjacent domains. Thus the problem comes down to whether the transition point of the two patterns conforming with the experimental result can be obtained by the Helfrich elastic curvature energy theory if one performs a more precise calculation. Our calculation is performed with the direct minimization method, with the two boundaries of domains constrained in two parallel planes, this is an effective method to guarantee the smoothness of the boundary. To allow the vesicle to have a sufficient freedom to evolve, only constraints of fixed reduced volume and area fraction are imposed (The usual implementation method of constraints with the enclosed volume and the area of each phase fixed is not appropriate in this case. It does not allow the vesicle to have enough freedom to evolve, since the two boundaries are constrained in two preassigned planes). For the experimental parameters of = 50 and = 0.02, the transition point for the two patterns is obtained to be o* = 0.49, which is quite close to the experimental result of o* = 0.5. In order to understand the budding process in the second process, a detailed study is also made with the direct minimization method. It is found that the budding process can occur only for high enough value ( qslant 7.0) and permeable membrane (in other words, no constraint of reduced volume is exerted). One possible mechanism of the permeation is the temporary passage caused by the defect in the bilayer membrane due to large reduced line tension, which needs to be further checked experimentally. The three-domain vesicles found in the experiment have rotational symmetry in the case of small (or large v). What is more, they have a reflective symmetric plane perpendicular to the rotational symmetric axis, thus only vesicles with Dh symmetry are considered in this paper.