Egg-Shaped Curves (2007)

Summary

(If clicked, to be seen by expansion.) Fig.3 Relationship between the ratio of the minor to the major axes and the values of volume and surface area of egg shaped solid figure. Especially, the relationship between the ratio of the minor to the major axes and the values of volume and surface area of egg shaped solid figure is shown in Fig.3. In purpose to calculate the coordinates data for displaying the relationship as shown in Fig.3, a C++ program using Eqs.(15b) and (16) etc.is composed as given by . return 3. Expression of the Equations of the Egg Shaped Curves with the Use of the Intermediate Variable In this section, the equations of the egg-shaped-curves expressed with the use of the intermediate variable Mr. Tadao ITOU (another equation of an egg-shaped-curve given by him is shown in ). Such an expression will be much convenient to display egg-shaped-curve graphically by computer aided calculation. In the first, we add the constant value of to the right hand side of Eq.(4) as according to the displacement of the trajectory by the value of in the direction as mentioned above. Then, . In the proceeding, substituting Eqs.(1), (2) ,(6) and (8) into the above equation, we obtain that . (17) In the next, substituting Eqs.(2) and (8) into Eq.(5), we obtain the following equation; . (18) In such a way, we obtained two equations describing an egg-shaped-curve with the use of an intermediate variable . The selection of the value of is not restricted under the condition that the continued range of is taken to be 2*pai. The above summary is described as the followings; , (19)where . It is noticed that the intermediate variable in Eq.(19) is defined in Fig.1 and then, does not indicate the phase angle of the egg curve in the polar coordinates. return 4. The Case Using Ellipsoid instead of Circle as a Basic Figure As seen in Fig.2, the basic figure of egg shaped curves is a circle in the above sections. If we adopt ellipse instead of circle as a basic figure, many other ty...