Egg-Shaped Curves

     For simple expression of the above equation, we displace the trajectory by the value of in the direction, and we introduce the next two constant values;
      ,    .              (8)
Then, Eq.(7) is rewritten into the following equation giving egg-like shaped curves.

      ,          (9)

where .
    If we solve Eq.(9) with the usual solution method of the 2nd order equation, we obtain that

      .          (9b)
Such the solved equation may be used for programming and calculation with computer.

     If we introduce , Eq.(9) is reduced into the following equation.

      .                                                (10)

    Thus, Eq.(10) takes the simple and beautiful form like as an egg-shaped curve.

    It is prefer that Eq.(10) is taken as the standard equation of egg-shaped curve.    However, as Eq.(9) has been treated from the beginning of this study, we intend to use Eq.(9) instead of Eq.(10) in below.

    In the case of , if we calculate Eq.(9) as varying the several values of with the use of computer, the each egg-like shaped curve is drawn as shown in Fig.2.    In the case of , the curve becomes a circle.    As increasing the value of , the more the curve is going to become near the shape of an actual egg,     In the case of , the curve approaches the shape of an actual egg most.    In this condition, the comparison between the egg shaped curve and the shape of an actual egg is shown in Fig.2a.

     Fig.1 and the culcuration process from Eqs.(1) to (6) may suggest the generation process of an actual chicken egg.

    The curve only in the case of becomes pointed at x=0.    In the cases except for , the curves are never pointed, as it is simply verified that the gradient of each curve at x=0 is infinitive, i.e., (dy/dx)x=0 = nfinitive.

   Mr. Akira NAGASHIMA indicated Fig.2 as an animation clearly in March in 2011.    I'll be thankful.    It is shown in Fig.2b.