Heart Curve III

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Heart Curve III

Nobuo YAMAMOTO

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Equation of heart curve is tried to be composed with the use of the exponential function
in the same way as described in the page of the equation of a curve expressing the shape of a letter S.

1. Method1

    The following two equations are supposed to be able to display a heart curve as the value of the intermediate variable varies in the range that .

                    ,                      (1)
and

                     ,                      (2)
where .
    When the intermediate variable varies in the range of , the curves which Eqs.(1) and (2) display are shown in the following figures.


Fig.1	Fig.2	Fig.3


Fig.4	Fig.5	Fig.6


Fig.7	Fig.8	Fig.9


Fig.10	Fig.11	Fig.12

When the above figures are painted, these are shown in the followings.

In purpose to calculate the numerical coordinates data of a heart curve as shown in Figs.1 to 12, a C++ program originated from Eqs.(1) and (2) is given by . By executing the either C++ program, a text file named "heart_curve3.txt" including the calculated data is produced. Each interval of these data is divided by 'comma'. After moving these calculated data into an Excel file, we obtain a heart curve with the use of a graph wizard attached on the Excel file.

2. Method 2

    The following two equations are considered corresponding to Eqs.(1) and (2).

                     ,                      (3)
and

                     ,                       (4)
where .
    When the intermediate variable varies in the range of , the curves which Eqs.(3) and (4) display are shown in the following figures.


Fig.13	Fig.14	Fig.15


Fig.16	Fig.17	Fig.18


Fig.19	Fig.20	Fig.21


Fig.22	Fig.23	Fig.24

When the above figures are painted, these are shown in the followings.

In purpose to calculate the numerical coordinates data of a heart curve as shown in Figs.13 to 24, a C++ program originated from Eqs.(3) and (4) is given by .

3. Method 3

    The following two equations are considered corresponding to Eqs.(1) and (2).

                     ,                       (5)
and

                     ,                       (6)
where .
    When the intermediate variable varies in the range of , the curves which Eqs.(5) and (6) display, and these painted ones are shown in the following figures.


Fig.25	Fig.26	Fig.27

In purpose to calculate the numerical coordinates data of a heart curve as shown in Figs.25 to 27, a C++ program originated from Eqs.(5) and (6) is given by .

4. Method 4

    The following two equations are considered corresponding to Eqs.(1) and (2).

                    ,                      (7)
and

                     ,                       (8)
where .
    When the intermediate variable varies in the range of , the curves which Eqs.(7) and (8) display are shown in the following figures.


Fig.28	Fig.29	Fig.30	Fig.31

When the above figures are painted, these are shown in the followings.

In purpose to calculate the numerical coordinates data of a heart curve as shown in Figs.28 to 31, a C++ program originated from Eqs.(7) and (8) is given by .

5. Method 5

    The following two equations are considered corresponding to Eqs.(1) and (2).

                    ,                      (9)
and

                     ,                      (10)
where .
    When the intermediate variable varies in the range of , the curves which Eqs.(9) and (10) display are shown in the following figures.


Fig.32	Fig.33	Fig.34	Fig.35

When the above figures are painted, these are shown in the followings.

In purpose to calculate the numerical coordinates data of a heart curve as shown in Figs.32 to 35, a C++ program originated from Eqs.(9) and (10) is given by .

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Updated: 2010.03.06, edited by N. Yamamoto.
Revised on Mar. 16, 2015, Jul. 24, 2016, May 04, 2020, Jan. 19, 2021, May 08, 2021 and Feb. 28, 2022.