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Eye Shaped Curves
Nobuo YAMAMOTO
1.  Expression by the 2nd order exponential function

We treat the equation given below.
,                       (1a)
where b > 0 and c > 0.    The solution of the above equation is obtained as
,                       (2a)
where the following condition must be needed only in the case of a < 0.
.                       (3a)
If we calculate Eq.(2a) numerically with the use of computer in the case of b=c=1, eye shaped curves are drawn as shown in Fig.1a.

 A C++ program, with which the numerical calculation of Eq.(2a) is executed to obtain the curves shown in Fig.1a, is given by .

In the next, the equation which expresses the both eyes with the use of two figures of Fig.1a is described as
,                       (1b)
where b > 0 and c > 0.    The solution of the above equation is obtained as
,                       (2b)
where the following condition must be needed only in the case of a < 0.
.                       (3b)
If we calculate Eq.(2b) numerically with the use of computer, curves of both eyes are drawn as shown in the following figures.

 (If clicked, it will be expanded.) Fig.1b Fig.1c Fig.1d Fig.1e Fig.1f Fig.1g Fig.1h Fig.1i

An animation of the above figures is as follows.

 A C++ program, with which the numerical calculation of Eq.(2b) is executed to obtain the curves of both eyes, is given by .

2.  Expression by the 2nd order exponential function of y and the 4th order exponential function of x

We treat the equation given below.
,                       (4)
where b > 0 and c > 0.    The solution of the above equation is obtained as
,                       (5)
where the following condition must be needed only in the case of a < 0.
.                       (6)
If we calculate Eq.(5) numerically with the use of computer in the case of b=c=1, curves are drawn as shown in Fig.2.

 A C++ program, with which the numerical calculation of Eq.(5) is executed to obtain the curves shown in Fig.2, is given by .

3.  Expression by the 4th order exponential function

We treat the equation given below.
,                       (7)
where b > 0 and c > 0.    The solution of the above equation is obtained as
,                       (8)
where the condition of Eq.(6) must be needed only in the case of a < 0.      If we calculate Eq.(8) numerically with the use of computer in the case of b=c=1, eye shaped curves are drawn as shown in Fig.3.

 A C++ program, with which the numerical calculation of Eq.(8) is executed to obtain the curves shown in Fig.3, is given by .

4.  Expression by the 4th order exponential function of y and the 2nd order exponential function of x

We treat the equation given below.
,                       (9)
where b > 0 and c > 0.    The solution of the above equation is obtained as
,                       (10)
where the condition of Eq.(3) must be needed only in the case of a < 0.      If we calculate Eq.(10) numerically with the use of computer in the case of b=c=1, curves are drawn as shown in Fig.4.

 A C++ program, with which the numerical calculation of Eq.(10) is executed to obtain the curves shown in Fig.4, is given by .

5.  The case with the use of the ordinary 2nd order equation

We treat the equation given below.
,                       (11)
where b > 0 and c > 0.    The solution of the above equation is obtained as
,                       (12)
where the following condition must be needed only in the case of a < 0.
.                       (13)
If we calculate Eq.(12) numerically with the use of computer in the case of b=0.8 and c=1, curves are drawn as shown in Fig.5.

 A C++ program, with which the numerical calculation of Eq.(12) is executed to obtain the curves shown in Fig.5, is given by .

6.  The case with the use of the ordinary 4nd order equation

We treat the equation given below.
,                       (14)
where b > 0 and c > 0.    The solution of the above equation is obtained as
,                       (15)
where the following condition must be needed only in the case of a < 0.
.                       (16)
If we calculate Eq.(15) numerically with the use of computer in the case of b=c=1, curves are drawn as shown in Fig.6.

 A C++ program, with which the numerical calculation of Eq.(15) is executed to obtain the curves shown in Fig.6, is given by .

7.  The case with the use of the 1st order hyper co-sinusoidal function

We treat the equation given below.
,                       (17)
where b > 0 and c > 0.    The solution of the above equation is obtained as
,                       (18)
where the following condition must be needed only in the case of a < 0.
.                       (19)
If we calculate Eq.(18) numerically with the use of computer in the case of b=c=1, curves are drawn as shown in Fig.7.

 A C++ program, with which the numerical calculation of Eq.(18) is executed to obtain the curves shown in Fig.7, is given by

8.  The case with the use of the 2nd order hyper co-sinusoidal function

We treat the equation given below.
,                       (20)
where b > 0 and c > 0.    The solution of the above equation is obtained as
,                       (21)
where the following condition must be needed only in the case of a < 0.
.                       (22)
If we calculate Eq.(21) numerically with the use of computer in the case of b=c=1, curves are drawn as shown in Fig.8.

 A C++ program, with which the numerical calculation of Eq.(21) is executed to obtain the curves shown in Fig.8, is given by

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Updated: 2008.09.11, edited by N. Yamamoto.
Revised on Jan. 03, 2014, Jan. 17, 2014, Jan. 23, 2014, Feb. 03, 2014, Jun. 10, 2014, Mar. 16, 2015, May 04, 2020, May 17, 2020, May 20, 2020, May 22, 2020, May 30, 2020, Jan. 27, 2021, May 08, 2021 and Mar. 22, 2022.