## 1 Introduction  
Bubbles floating on a liquid can be seen in various places, and an understanding of the related phenomena is important in the natural sciences and industrial fields. For example, the bursting of floating bubbles in the ocean affects atmospheric circulation and marine pollution [1; 2; 3; 4]. In a nuclear power plant accident, the bursting of bubbles in the reactor acts as a source of radionuclides [5; 6]. Volcanic eruptions are caused by bubbles with sizes measured in meters [7]. In food engineering, bubbles are involved in the release of foams and flavors, e.g., champagne, espresso, and beer [8; 9; 10; 11]. Recently, boiling heat transfer in a complex nanofluid is closely related to bubble nucleation and growth [12; 13]. Bubbles generated in a liquid rise due to buoyancy, and then burst after reaching the liquid surface [14]. Bubbles floating on a liquid can remain in quasi-equilibrium for some time before bursting. During this time, the thin liquid film that defines the surface of the bubble gradually becomes thinner and unstable due to gravity, external disturbance, and impurities [15]. After bursting, the dynamics of the bubble cannot be simulated by the present numerical method, and the partial differential equations of the conservation laws must be solved using the computational fluid dynamics techniques such as the finite element method [16], the arbitrary Lagrangian-Eulerian moving mesh method [17], the two-fluid two-phase flow model [18; 19], and the adaptive multi-phase method [20]. As shown schematically in Figure 1, the shape of the floating bubbles in the quasi-equilibrium state can be divided into three curves, namely the cap, the cavity, and the meniscus. These three curves meet at a junction point \(c\). As the various dynamic phenomena that occur when a floating bubble bursts depend largely on its initial shape, the accurate prediction of this shape is important [21; 22; 23; 24; 25; 26; 27]. If the thickness and drainage of the liquid film can be neglected, then the shape of the bubble in the quasi-equilibrium state is determined by the Young-Laplace equation representing the balance between the forces of hydrostatic pressure and capillary action [28; 29; 30; 31; 32]. Floating bubbles vary in shape from spherical to hemispherical, depending on their size. Therefore, the Bond number, \(Bo\), is used as a key parameter. [PERSON] and [PERSON] [28] presented a systematic theoretical equation for hydrostatic pressure and capillary forces and obtained a series solution of drops of fluids. Subsequently, [PERSON] [29] calculated the shapes of floating bubbles by using a semi-graphical method. At the same time, [PERSON] [30] used the Table of Bashforth and [PERSON] [28] to find the junction point and calculate the shapes of bubbles of various sizes. [PERSON] and [PERSON] [31] numerically calculated the bubble shape for a broader range of the bubble sizes (which corresponds to a range of \(0.003<Bo<241\), computed based on the parameter \(B\) values of [31]), and derived an analytic solution using a perturbation technique for tiny bubbles that are difficult to calculate numerically. Recently, [PERSON] and [PERSON] [23] obtained the bubble shape via a method similar to that of [PERSON] [29]; however, there is no detailed information on their numerical method and on the accuracy of their solution. [PERSON] [32] used a similar method to [PERSON] [30] and presented the shapes of bubbles for Bond numbers between 0.1 to 100. [PERSON] et al. [33] developed a model considering the film weight of giant soap bubbles and compared the results of numerical solutions and experiments. [PERSON] and [PERSON] [34] studied the coalescence of floating bubbles for a wide range of Bond number, and characterized the evolution of the underwater neck and the surface bridge. [PERSON] et al. [35] investigated the life of a floating bubble by exploring the role of drainage and evaporation on film thinning.