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Heart Curve II

Nobuo YAMAMOTO

You can copy and use all the figures in this page freely.

1.  Preface

    Many types of heart curves are seen on the sites of  Heart Curve - Mathematische Basteleien,  First heart Curve - Wolfram|Alpha and Heart Curve -- from Wolfram MathWorld as representations.
    In this page, it is tried that a Cardioid is reformed into a heart curve.
    The first method is to make a corner on the round bottom of a Cardioid by decreasing the phase angle linearly.    The second method is to make a corner on the round bottom by a nonlinear conversion of the phase angle.


Fig.1  Cardioid
2.   Method 1

    Though the Cardioid is introduced in the page of  Wolfram Math World,  the equation expressing a Cardioid is rewritten as the following after the length and the width are replaced.
          ,                      (1)
where and indicate the moving radius and the phase angle respectively.
    The location of the coordinate origin will be intended to be replaced to the bottom of the Cardioid in Fig.1.    In this figure,
          .
          ,                      (2)
where .    Moreover,

          .
    If we substitute Eqs.(1) and (2) into the above equation, we obtain

Fig.2  The definition of
          ,                      (3)
where .

    The newly defined phase angle of the Cardioid after the replacement of the coordinate origin, which is shown in Fig.1, is written as
          ,                      (4)
where .


Fig.3  Conversion of the phase angle
from into


    In the next, in order that the bottom of the Cardioid is reformed into a heart curve with a corner having the desired angle (as seen in Fig.2), we converse the phase angle of the Cardioid into the newly defined phase angle of the heart curve linearly as shown in Fig.3.    A conversion equation which satisfies the above mention may be given as
          .                      (5)
    If we substitute Eq.(4) into Eq.(5), the conversion equation from to is obtained as
          .                      (6)

    The orthogonal coordinate expression of the heart curve may be written as the following two equations;
          .                      (7)
          ,                      (8)
where indicates compression rate in the length direction.    When this rate does not exist, a stretched heart curve may appear.

    By calculating Eqs.(2), (3), (6), (7) and (8), the coordinate data of the heart curve are obtained.    Examples of such obtained curves in the case of are shown in Figs.4 to 8 where decides only the size and does not relate to the shape.

Fig.4
Fig.5

Fig.6a
Fig.6b  =30°, b=20%

Fig.6c  =30°, b=23%
Fig.7a  =45°, b=25%

Fig.7b  =45°, b=30%
Fig.7c  =45°, b=35%

Fig.8a  =60°, b=35%
Fig.8b  =60°, b=40%

Fig.8c  =60°, b=45%
Fig.9a  =90°, b=50%

Fig.9b
Fig.10  =120°, b=80%

    When the above figures are painted, these are shown in the followings.
Fig.4 α=5°, b=3%Fig.5 α=15°, b=10%Fig.6a α=30°, b=18%Fig.6b α=30°, b=20%Fig.6c α=30°, b=23%Fig.7a α=45°, b=25%Fig.7b α=45°, b=30%Fig.7c α=45°, b=35%Fig.8a α=60°, b=35% Fig.9b α=90゜, b=55%Fig.10 α=120°, b=80% Fig.8b α=60°, b=40%Fig.8c α=60°, b=45%Fig.9a α=90°, b=50%





    In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(2), (3), (6), (7) and (8), a C++ program is given by C++_program I.
    By executing the C++ program, a text file named "heart_curve_a.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.


3.    Method 2
     In this section, we try to make a corner on the bottom of Cardioid by a nonlinear conversion of the phase angle and try to bring the Cardioid into the shape of a heart curve.    Specifically, we converse the phase angle of a Cardioid (in Fig.11a) into the phase angle of a heart curve (in Fig.11b) in the manner as shown in Fig.11c.    If the conversion function is given as , the conversion relation is expressed as
          .                      (9a)
    Here, we adopt the following equation using a linear combination of the square root function and the proportional relation , which satisfies the characteristics given in Fig.11c.
          ,                      (9b)
where is the positive real number.     may be regarded as a strength of the conversion.    It has to be taken in account that has not periodicity and is applied only in the range of as shown in Fig.11c.
    Thus, we can make a corner on the bottom of the Cardioid.
    Further, a little different type of heart curves are described in Heart curves IIb in the case that Fig.11c is expressed by using inverse-trigonometrical function instead of using the square root function of Eq.(9b).

Fig.11a  Cardioid
Fig.11b  Heart curve

Fig.11c  Conversion of the phase angle
from into

    Moreover, in order to obtain beautiful shape of heart figure, both of the coefficient for the reformation and the compression coefficient in the length direction are included in the following conversion equations in the coordinates.
          ,                        (10a)
and
          ,                      (10b)
where
          ,                      (11)

    By calculating Eqs.(9a), (9b), (10a), (10b) and (11), the coordinate data of a heart curve are obtained.    Examples of such obtained curves in the case of are shown in Figs.12 to 24 where decides only the size and does not relate to the shape.


Fig.12a  b=1, c=0.3, d=1
Fig.12b  b=1, c=0.4, d=1

Fig.12c  b=1, c=0.4, d=1.1
Fig.13a  b=1.2, c=0.4, d=1

Fig.13b  b=1.2, c=0.5, d=1.1
Fig.13c  b=1.2, c=0.6, d=1.2

Fig.14a  b=1.5, c=0.5, d=0.95
Fig.14b  b=1.5, c=0.5, d=1

Fig.14c  b=1.5, c=0.6, d=1
Fig.14d  b=1.5, c=0.6, d=1.05

Fig.14e  b=1.5, c=0.6, d=1.1
Fig.14f  b=1.5, c=0.6, d=1.15

Fig.14g  b=1.5, c=0.6, d=1.2
Fig.14h  b=1.5, c=0.65, d=1

Fig.14i  b=1.5, c=0.65, d=1.1
Fig.14j  b=1.5, c=0.65, d=1.2

Fig.14k  b=1.5, c=0.65, d=1.3
Fig.14l  b=1.5, c=0.7, d=1

Fig.14m  b=1.5, c=0.7, d=1.2
Fig.14n  b=1.5, c=0.7, d=1.4

Fig.15a  b=1.65, c=0.6, d=1
Fig.15b  b=1.65, c=0.6, d=1.1

Fig.16a  b=1.8, c=0.3, d=0.7
Fig.16b  b=1.8, c=0.3, d=0.8

Fig.17a  b=1.8, c=0.5, d=0.8
Fig.17b  b=1.8, c=0.5, d=0.9

Fig.17c  b=1.8, c=0.5, d=1
Fig.18a  b=1.8, c=0.6, d=1

Fig.18b  b=1.8, c=0.6, d=1.1
Fig.19a  b=1.8, c=0.67, d=1

Fig.19b  b=1.8, c=0.67, d=1.05
Fig.19c  b=1.8, c=0.67, d=1.1

Fig.19d  b=1.8, c=0.67, d=1.15
Fig.19e  b=1.8, c=0.67, d=1.2

Fig.19f  b=1.8, c=0.7, d=1
Fig.19g  b=1.8, c=0.7, d=1.05

Fig.19h  b=1.8, c=0.7, d=1.1
Fig.19i  b=1.8, c=0.7, d=1.15

Fig.19j  b=1.8, c=0.7, d=1.2
Fig.19k  b=1.8, c=0.7, d=1.25

Fig.19l  b=1.8, c=0.7, d=1.3
Fig.19m  b=1.8, c=0.7, d=1.35

Fig.19n  b=1.8, c=0.7, d=1.4
Fig.20a  b=2, c=0.3, d=0.7

Fig.20b  b=2, c=0.3, d=0.8
Fig.20c  b=2, c=0.5, d=0.7

Fig.20d  b=2, c=0.5, d=0.8
Fig.21a  b=2, c=0.6, d=0.9

Fig.21b  b=2, c=0.65, d=1
Fig.21c  b=2, c=0.7, d=1

Fig.21d  b=2, c=0.7, d=1.05
Fig.21e  b=2, c=0.7, d=1.1

Fig.21f  b=2, c=0.7, d=1.15
Fig.21g  b=2, c=0.7, d=1.2

Fig.21h  b=2, c=0.7, d=1.25
Fig.21i  b=2, c=0.7, d=1.3

Fig.21j  b=2, c=0.7, d=1.4
Fig.21k  b=2, c=0.73, d=1

Fig.21l  b=2, c=0.73, d=1.1
Fig.21m  b=2, c=0.73, d=1.2

Fig.21n  b=2, c=0.73, d=1.3
Fig.21o  b=2, c=0.73, d=1.4

Fig.22a  b=2.2, c=0.7, d=0.8
Fig.22b  b=2.2, c=0.7, d=0.9

Fig.22c  b=2.2, c=0.7, d=1
Fig.22d  b=2.2, c=0.7, d=1.05

Fig.22e  b=2.2, c=0.7, d=1.1
Fig.22f  b=2.2, c=0.7, d=1.15

Fig.22g  b=2.2, c=0.75, d=1
Fig.22h  b=2.2, c=0.75, d=1.1

Fig.22i  b=2.2, c=0.75, d=1.2
Fig.22j  b=2.2, c=0.75, d=1.25

Fig.22k  b=2.2, c=0.75, d=1.3
Fig.23a  b=2.5, c=0.7, d=0.8

Fig.23b  b=2.5, c=0.78, d=1
Fig.23c  b=2.5, c=0.78, d=1.1

Fig.23d  b=2.5, c=0.78, d=1.2
Fig.23e  b=2.7, c=0.78, d=0.95

Fig.23f  b=2.7, c=0.78, d=1
Fig.23g  b=2.7, c=0.78, d=1.05

Fig.23h  b=2.7, c=0.82, d=1
Fig.23i  b=2.7, c=0.82, d=1.1

Fig.23j  b=2.7, c=0.82, d=1.2


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

Fig.12a b=1, c=0.3, d=1Fig.12b b=1, c=0.4, d=1Fig.12c b=1, c=0.4, d=1.1Fig.13a b=1.2, c=0.4, d=1Fig.13b b=1.2, c=0.5, d=1.1Fig.13c b=1.2, c=0.6, d=1.2Fig.14a b=1.5, c=0.5, d=0.95Fig.14b b=1.5, c=0.5, d=1Fig.14c b=1.5, c=0.6, d=1Fig.14d b=1.5, c=0.6, d=1.05Fig.14e b=1.5, c=0.6, d=1.1Fig.14f b=1.5, c=0.6, d=1.15Fig.14g b=1.5, c=0.6, d=1.2Fig.14h b=1.5, c=0.65, d=1Fig.14i b=1.5, c=0.65, d=1.1Fig.14j b=1.5, c=0.65, d=1.2Fig.14k b=1.5, c=0.65, d=1.3Fig.14l b=1.5, c=0.7, d=1Fig.14m b=1.5, c=0.7, d=1.2図14n b=1.5, c=0.7, d=1.4Fig.15a b=1.65, c=0.6, d=1Fig.15b b=1.65, c=0.6, d=1.1Fig.16a b=1.8, c=0.3, d=0.7Fig.16b b=1.8, c=0.3, d=0.8Fig.17a b=1.8, c=0.5, d=0.8Fig.17b b=1.8, c=0.5, d=0.9Fig.17c b=1.8, c=0.5, d=1Fig.18a b=1.8, c=0.6, d=1Fig.18b b=1.8, c=0.6, d=1.1Fig.19a b=1.8, c=0.67, d=1Fig.19b b=1.8, c=0.67, d=1.05Fig.19c b=1.8, c=0.67, d=1.1Fig.19d b=1.8, c=0.67, d=1.15Fig.19e b=1.8, c=0.67, d=1.2Fig.19f b=1.8, c=0.7, d=1Fig.19g b=1.8, c=0.7, d=1.05Fig.19h b=1.8, c=0.7, d=1.1Fig.19i b=1.8, c=0.7, d=1.15Fig.19j b=1.8, c=0.7, d=1.2Fig.19k b=1.8, c=0.7, d=1.25Fig.19l b=1.8, c=0.7, d=1.3Fig.19m b=1.8, c=0.7, d=1.35Fig.19n b=1.8, c=0.7, d=1.4Fig.20a b=2, c=0.3, d=0.7Fig.20b b=2, c=0.3, d=0.8Fig.20c b=2, c=0.5, d=0.7Fig.20d b=2, c=0.5, d=0.8Fig.21a b=2, c=0.6, d=0.9Fig.21b b=2, c=0.65, d=1Fig.21c b=2, c=0.7, d=1Fig.21d b=2, c=0.7, d=1.05Fig.21e b=2, c=0.7, d=1.1Fig.21f c=0.7, d=1.15Fig.21g b=2, c=0.7, d=1.2Fig.21h b=2, c=0.7, d=1.25Fig.21i b=2, c=0.7, d=1.3Fig.21j b=2, c=0.7, d=1.4Fig.21k b=2, c=0.73, d=1Fig.21l b=2, c=0.73, d=1.1Fig.21m b=2, c=0.73, d=1.2Fig.21n b=2, c=0.73, d=1.3Fig.21o b=2, c=0.73, d=1.4Fig.22a b=2.2, c=0.7, d=0.8Fig.22b b=2.2, c=0.7, d=0.9Fig.22c b=2.2, c=0.7, d=1Fig.22d b=2.2, c=0.7, d=1.05Fig.22e b=2.2, c=0.7, d=1.1Fig.22f b=2.2, c=0.7, d=1.15Fig.22g b=2.2, c=0.75, d=1Fig.22h b=2.2, c=0.75, d=1Fig.22i b=2.2, c=0.75, d=1Fig.22j b=2.2, c=0.75, d=1.25Fig.22k b=2.2, c=0.75, d=1.3Fig.23a b=2.5, c=0.7, d=0.8Fig.23b b=2.5, c=0.78, d=1Fig.23c b=2.5, c=0.78, d=1.1Fig.23d b=2.5, c=0.78, d=1.2Fig.23e b=2.7, c=0.78, d=0.95Fig.23f b=2.7, c=0.78, d=1Fig.23g b=2.7, c=0.78, d=1.05Fig.23h b=2.7, c=0.82, d=1Fig.23i b=2.7, c=0.82, d=1.1Fig.23j b=2.7, c=0.82, d=1.2

    In another method, the better shaped heart curves are obtained and shown in Heart curve IIb of "Heart Curves IIb".

    When we gather all the data of parameters with the use of which many heart curves have been displayed as above, we come to recognize the region of the parameters , and in which a heart curve may be found as seen in Fig.24.

In ( , ) plain
In ( , ) plain
Fig.24   The region of the parameters , and
in which a heart curve may be found
Herein, pink colored area represents its region, and
blue colored dots represents data points which are
obtained when heart curves are displayed as above.


    In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(9a), (9b), (10a), (10b) and (11), a C++ program is given by C++_program II.
    By executing the C++ program, a text file named "heart_curve_b.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 heart a curve with the use of a graph wizard attached on the excel file.


   The other types of curves besides the heart shaped ones are also obtained as follows.

Fig.25a
Fig.25b

Fig.25c
Fig.25d

Fig.25e
Fig.25f

   When the above figures are painted, these are shown in the followings.
Fig.25a b=2.5, c=1, d=1.2Fig.25b b=2.5, c=0.95, d=1Fig.25c b=2.5, c=0.9, d=0.9Fig.25d b=2.5, c=0.9, d=1Fig.25e b=2.5, c=1.1, d=1.1Fig.25f b=2.5, c=2, d=1


4.   Method 3

    We try to make the dent of a heart figure by the method 2 deeper and wider.    To do this, Eq.(10b) is changed to the following equation besides Eq.(10a) remains unchanged.
          ,                      (12)
where is given by Eq.(10), and is the newly introduced constant.

    By calculating Eqs.(9), (10a), (11) and (12), the coordinate data of a heart curve are obtained.    Examples of such obtained curves in the case of are shown in Figs.26 to 32, where Fig.26 is reformed from Fig.14b, Fig.27 from Fig.15, Fig.28 from Fig.18, Fig.29 from Figs.19b and 19c, Fig.30 from Fig.22e, Fig.31 from Fig.23c and Fig.32 from Fig.24c respectively.

Fig.26a
Fig.26b

Fig.26c
Fig.26d

Fig.27a
Fig.27b

Fig.27c
Fig.27d

Fig.28a
Fig.28b

Fig.28c
Fig.28d

Fig.29a
Fig.29b

Fig.29c
Fig.29d

Fig.30a
Fig.30b

Fig.31a
Fig.31b

Fig.32a
Fig.32b

     When the above figures are painted, these are shown in the followings.
Fig.26a b=1.5, c=0.5, d=1, p=0.1Fig.26b b=1.5, c=0.5, d=1, p=0.2Fig.26c b=1.5, c=0.5, d=1.1, p=0.1Fig.26d b=1.5, c=0.5, d=1.1, p=0.2Fig.27a b=1.65, c=0.6, d=1, p=0.1Fig.27b b=1.65, c=0.6, d=1, p=0.2Fig.27c b=1.65, c=0.6, d=1.1, p=0.1Fig.27d b=1.65, c=0.6, d=1.1, p=0.2Fig.28a b=1.8, c=0.6, d=0.95, p=0.1Fig.28b b=1.8, c=0.6, d=1, p=0.1Fig.28c b=1.8, c=0.6, d=1.05, p=0.1Fig.28d b=1.8, c=0.6, d=1, p=0.2Fig.29a b=1.8, c=0.7, d=1.1, p=0.1Fig.29b b=1.8, c=0.7, d=1.1, p=0.2Fig.29c b=1.8, c=0.7, d=1.1, p=0.2Fig.29d b=1.8, c=0.7, d=1.2, p=0.2Fig.30a b=2, c=0.7, d=1.1, p=0.1Fig.30b b=2, c=0.7, d=1.1, p=0.2Fig.31a b=2.2, c=0.7, d=1, p=0.1Fig.31b b=2.2, c=0.7, d=1, p=0.2Fig.32a b=2.5, c=0.78, d=1.1, p=0.1Fig.32b b=2.5, c=0.78, d=1.1, p=0.2


    In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(9), (10a), (11) and (12), a C++ program is given by C++_program III.
    By executing the C++ program, a text file named "heart_curve_c.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 heart a curve with the use of a graph wizard attached on the excel file.


5.   Method 4

    According to the concept as like as mentioned in the previous method 3 again, Eq.(10b) is changed to the following equation besides Eq.(10a) remains unchanged.
          ,                      (13)
where is given by Eq.(10a), and is the newly introduced constant.

    By calculating Eqs.(9), (10a), (11) and (13), the coordinate data of a heart curve are obtained.    Examples of such obtained curves in the case of are shown in Figs.33 to 39, where Fig.33 is reformed from Fig.14b, Fig.34 from Fig.15, Fig.35 from Fig.18, Fig.36 from Figs.19b and 19c, Fig.37 from Fig.22e, Fig.38 from Fig.23c and Fig.39 from Fig.24c respectively.

Fig.33a
Fig.33b

Fig.34a
Fig.34b

Fig.34c
Fig.34d

Fig.35a
Fig.35b

Fig.35c
Fig.35d

Fig.36a
Fig.36b

Fig.36c
Fig.36d

Fig.37a
Fig.37b

Fig.38a
Fig.38b

Fig.39a
Fig.39b

     When the above figures are painted, these are shown in the followings.
Fig.33a b=1.5, c=0.5, d=1, p=0.1Fig.33b b=1.5, c=0.5, d=1, p=0.2Fig.34a b=1.65, c=0.6, d=1, p=0.1Fig.34b b=1.65, c=0.6, d=1, p=0.2Fig.34c b=1.65, c=0.6, d=1.1, p=0.1Fig.34d b=1.65, c=0.6, d=1.1, p=0.2Fig.35a b=1.8, c=0.6, d=0.95, p=0.1Fig.35b b=1.8, c=0.6, d=1, p=0.1Fig.35c b=1.8, c=0.6, d=1.05, p=0.1Fig.35d b=1.8, c=0.6, d=1, p=0.2Fig.36a b=1.8, c=0.7, d=1.1, p=0.1Fig.36b b=1.8, c=0.7, d=1.1, p=0.1Fig.36c b=1.8, c=0.7, d=1.2, p=0.1Fig.36d b=1.8, c=0.7, d=1.2, p=0.2Fig.37a b=2, c=0.7, d=1.1, p=0.1Fig.37b b=2, c=0.7, d=1.1, p=0.2Fig.38a b=2.2, c=0.7, d=1, p=0.1Fig.38b b=2.2, c=0.7, d=1, p=0.2Fig.39a b=2.2, c=0.78, d=1.1, p=0.1Fig.39b b=2.2, c=0.78, d=1.1, p=0.2



    In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.Eqs.(9), (10a), (11) and (13), a C++ program is given by C++_program IV.
    By executing the C++ program, a text file named "heart_curve_d.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 heart a curve with the use of a graph wizard attached on the excel file.


6.   Method 5

     We try to make the dent of a heart figure by the method 1 deeper and wider.    To do this, Eq.(8) is changed to the following equation besides Eq.(7) remains unchanged.
          ,                      (14)
where is given by Eq.(7), and is the newly introduced constant.

    By calculating Eqs.(2), (3), (6), (7) and (14), the coordinate data of a heart curve are obtained.    Examples of such obtained curves in the case of are shown in Figs.40 to 45, where Fig.40 is reformed from Fig.4, Fig.41 from Fig.6b, Fig.42 from Fig.7b, Fig.43 from Figs.8, Fig.44 from Fig.9b and Fig.45 from Fig.10 respectively.

Fig.40
Fig.41

Fig.42
Fig.43a

Fig.43b
Fig.44

Fig.45

     When the above figures are painted, these are shown in the followings.
Fig.40 α=5°, b=3%, p=0.1Fig.41 α=30°, b=20%, p=0.1Fig.42 α=45°, b=30%, p=0.1Fig.43a α=60°, b=37%, p=0.1Fig.43b α=60°, b=3%, p=0.1Fig.44 α=90°, b=55%, p=0.1Fig.45 α=120°, b=80%, p=0.2

    In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(2), (3), (6), (7) and (14), a C++ program is given by C++_program V.
    By executing the C++ program, a text file named "heart_curve_e.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 heart a curve with the use of a graph wizard attached on the excel file.


7.    Method 6

    According to the concept as like as mentioned in the previous method 5 again, Eq.(8) is changed to the following equation besides Eq.(7) remains unchanged.
          ,                      (15)
where is given by Eq.(7), and is the newly introduced constant.

    By calculating Eqs.(2), (3), (6), (7) and (15), the coordinate data of a heart curve are obtained.    Examples of such obtained curves in the case of are shown in Figs.46 to 51, where Fig.46 is reformed from Fig.4, Fig.47 from Fig.6b, Fig.48 from Fig.7b, Fig.49 from Figs.8, Fig.50 from Fig.9b and Fig.51 from Fig.10 respectively.

Fig.46
Fig.47

Fig.48
Fig.49a

Fig.49b
Fig.50a

Fig.50b
Fig.51a

Fig.51b
Fig.51c

     When the above figures are painted, these are shown in the followings.
Fig.46 α=5°, b=3%, p=0.02Fig.47 α=20°, b=20%, p=0.03Fig.48 α=45°, b=30%, p=0.03Fig.49a α=60°, b=37%, p=0.05Fig.49b α=60°, b=37%, p=0.18Fig.50a α=90°, b=55%, p=0.08Fig.50b α=90°, b=55%, p=0.18Fig.51a α=120°, b=80%, p=0.15Fig.51b α=120°, b=80%, p=0.3Fig.51c α=120°, b=75%, p=0.25



    In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(2), (3), (6), (7) and (15), a C++ program is given by C++_program VI.
    By executing the C++ program, a text file named "heart_curve_f.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 heart a curve with the use of a graph wizard attached on the excel file.


8.    Method 7


Fig.52   The distorted Cardioid in which the moving
radius is multiplied by .
    We try to apply the concept of the reformation of heart curve with the introduction of the coefficient (as described in Eqs.(10a) and (10b) in the method 2) to the above method 5.    As the concept in the method 5 is based on the method 1, should be multiplied by .    Then, the following equation into which Eq.(1) is rewritten is corresponding to Eqs.(10a) and (10b).
          ,                      (16)
where denotes the phase angle as shown in Fig.52.
    As a result of such the procedure, Fig.1 goes to Fig.51 in which the length of the moving radius in the length direction changes from to .
    Therefore, we must use the following equation, which is led from the equation situated immediately on Eq.(2), instead of Eq.(2).
          .                      (17)
Moreover, we must use the following equation, which is led from the equation situated immediately on Eq.(3), instead of Eq.(3).
          .                      (18)
    On the other hand, Eqs.(4), (5), (6) and (7) and Figs.2 and 3 are all applied also in this section.    However, we must use Eq.(14) instead of Eq.(8).

    By calculating Eqs.(16), (17), (18), (6), (7) and (14), the coordinate data of a heart curve are obtained.    Examples of such obtained curves in the case of are shown in Figs.53 to 54, where Fig.53 and Fig.54 are reformed from Fig.40 and Fig.43 respectively.

Fig.53a
Fig.53b

Fig.53c
Fig.53d

Fig.54a
Fig.54b

Fig.54c
Fig.54d

    When the above figures are painted, these are shown in the followings.
Fig.53a α=5°, b=3.5%, c=0.2, p=0Fig.53b α=5°, b=3.5%, c=0.3, p=0Fig.53c α=5°, b=3.5%, c=0.1, p=0.1Fig.53d α=5°, b=3.5%, c=0.2, p=0.1Fig.54a α=60°, b=37%, c=0.2, p=0Fig.54b α=60°, b=45%, c=0.2, p=0Fig.54c α=60°, b=37%, c=0.1, p=0.1Fig.54d α=60°, b=45%, c=0.2, p=0.1



In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(16), (17), (18), (6), (7) and (14), a C++ program is given by C++_program VII.
    By executing the C++ program, a text file named "heart_curve_g.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 heart a curve with the use of a graph wizard attached on the excel file.


9.   Method 8

     We try to apply the concept of the reformation of heart curve with the introduction of the coefficient (as described in Eqs.(10a) and (10b) in the method 2) to the above method 6.    The consideration and the procedure are as the same as mentioned in the above method 7.    Although Figs.2, 3 and 51, and Eqs.(16), (17), (18), (4), (5), (6) and (7) can all be applied also in this section, only Eq.(15) must be used instead of Eq.(8).

    By calculating Eqs.(16), (17), (18), (6), (7) and (15), the coordinate data of a heart curve are obtained.    Examples of such obtained curves in the case of are shown in Figs.55 to 56, where Fig.55 and Fig.56 are reformed from Fig.46 and Fig.49 respectively.

Fig.55a
Fig.55b

Fig.55c
Fig.55d

Fig.56a
Fig.56b

Fig.56c
Fig.56d

     When the above figures are painted, these are shown in the followings.
Fig.55a α=5°, b=3.5%, c=0.1, p=0Fig.55b α=5°, b=3.5%, c=0.2, p=0Fig.55c α=5°, b=3.5%, c=0.1, p=0.02Fig.55d α=5゜, b=3%, c=0.1, p=0.04Fig.56a α=60°, b=37%, c=0.1, p=0Fig.56b α=60°, b=45%, c=0.2, p=0Fig.56c α=60°, b=37%, c=0.1, p=0.05Fig.56d α=60°, b=37%, c=0.1, p=0.1



In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(16), (17), (18), (6), (7) and (15), a C++ program is given by C++_program VIII.
    By executing the C++ program, a text file named "heart_curve_h.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 heart a curve with the use of a graph wizard attached on the excel file.



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Updated: 2009.02.16, edited by N. Yamamoto.
Revised in Apr. 23, 2010, Dec. 10, 2011, Aug. 05, 2012, Sep. 04, 2013, Jan. 09, 2014, Jan. 14, 2014, Mar. 16, 2015, Dec. 20, 2015, Jul. 22, 2016, May 05, 2020, May 17, 2020 and May 24, 2020.