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Heart Curve II b

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.
     Although it is tried that a Cardioid is reformed into a heart curve also in this page as samely as in the section 3 of the page of Heart Curve II (described in Heart Vurve II), the generalized treatment is performed in this page, and then, the conversion equation of Eq.(9b) in the section 3 of the page of Heart Curve II is displaced to the equation using inverse-sinusoidal function etc. from using the square root function.


2.   Fundamental properties


     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.

     Here, we try to make a corner on the bottom of the Cardioid so as to converse the phase angle of a Cardioid (in Fig.1) into the phase angle of a heart curve (in Fig.2) in the manner as shown in Fig.3.    Then, if the inverse-conversion function is given as , the following relation is expressed as
           .                      (2)
     Thus, we should make a corner on the bottom of the Cardioid.

Fig.1  Cardioid
Fig.2  Heart curve

Fig.3  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.
          ,                        (3a)
and
          .                      (3b)

3.    The case using the inverse-sinusoidal function as

     We adopt the following equation using a linear combination of the inverse-sinusoidal function and the proportional relation , which combination satisfies the characteristics given in Fig.3.
          ,                      (4)
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.3.
    In the result to use the above equation, a heart curve is expected to have a gentle cut on the heart.    However, sharpness under the heart may tend to flow lengthily.

     By calculating Eqs.(1), (2), (3a), (3b) and (4), the coordinate data of a heart curve are obtained.    Examples of such obtained curves in the case of are shown in Figs.4 to 19 where decides only the size and does not relate to the shape.

Fig.4
Fig.5a

Fig.5b
Fig.5c

Fig.6a
Fig.6b

Fig.6c
Fig.6d

Fig.6e
Fig.7a

Fig.7b
Fig.7c

Fig.7d
Fig.8a

Fig.8b
Fig.8c

Fig.8d
Fig.9a

Fig.9b
Fig.9c

Fig.9d
Fig.9e

Fig.10
Fig.11a

Fig.11b
Fig.11c

Fig.11d
Fig.12a

Fig.12b
Fig.12c

Fig.12d
Fig.12e

Fig.13a
Fig.13b

Fig.13c
Fig.13d

Fig.13e
Fig.14a

Fig.14b
Fig.14c

Fig.14d
Fig.14e

Fig.14f
Fig.14g

Fig.15a
Fig.15b

Fig.15c
Fig.15d

Fig.15e
Fig.15f

Fig.16a
Fig.16b

Fig.16c
Fig.17a

Fig.17b
Fig.17c

Fig.17d
Fig.17e

Fig.18a
Fig.18b

Fig.18c
Fig.18d

Fig.18e
Fig.18f

Fig.18g

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

Fig.4 Fig.5a Fig.5b Fig.5c Fig.6a Fig.6b Fig.6c Fig.6d Fig.6e Fig.7a Fig.7b Fig.7c Fig.7d Fig.8a Fig.8b Fig.8c Fig.8d Fig.9a Fig.9b Fig.9c Fig.9d Fig.9e Fig.10 Fig.11a Fig.11b Fig.11c Fig.11d Fig.12a 図12d Fig.12c Fig.12d Fig.12f Fig.13a Fig.13b Fig.13c Fig.13d Fig.13e Fig.14a Fig.14b Fig.14c Fig.14d Fig.14e Fig.14f Fig.14g Fig.15a Fig.15b Fig.15c Fig.15d Fig.15e Fig.15f Fig.16a Fig.16b Fig.16c Fig.17a Fig.17b Fig.17c Fig.17d Fig.17e Fig.18a Fig.18b Fig.18c Fig.19a Fig.19b Fig.19c Fig.19d

　

     In another method, slightly different shaped heart curves are obtained and shown in Heart curve II of "Heart Curves II".

     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.19.

In ( , ) plain
Fig.19a   The region of the parameters and in which
a heart curve may be found
Herein, pink colored area represents its region, and dark
pink colored dots represents data points which are obtained
when heart curves are displayed as above.


In ( , ) plain
Fig.19b   The region of the parameters and in which
a heart curve may be found
Herein, pink colored area represents its region, and dark
pink 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.(1), (2), (3a), (3b) and (4), a C++ program is given by C++_program.
     By executing the C++ program, a text file named "heart_curve_2b_1.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.20a
Fig.20b

Fig.20c
Fig.20d

Fig.20e

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

Fig.20a Fig.20b Fig.20c Fig.20d Fig.20e


4.    The case using the inverse-sinusoidal function with narrower period interval as


     In the previous section, sharpness under a heart tends to flow lengthily.    Here, this tendency will be corrected.
    Though the full range of one period interval of the inverse sinusoidal function was adopted in the previous section, the narrower period interval (where ) is adopted here.      Then, the following linear combination of the inverse-sinusoidal function with narrower period interval and the proportional relation is used instead of Eq.(4) written in the previous section.
          ,                      (5)
where .    When , Eq.(5) is led to Eq.(4).

     By calculating Eqs.(1), (2), (3a), (3b) and (5), the coordinate data of heart curves are obtained.      Examples of such obtained curves in the case of are shown in Figs.21 to 36 where decides only the size and does not relate to the shape.

Fig.21
Fig.22a

Fig.22b
Fig.22c

Fig.23a
Fig.23b

Fig.23c
Fig.23d

Fig.23e
Fig.23f

Fig.24a
Fig.24b

Fig.24c
Fig.24d

Fig.24e
Fig.25a

Fig.25b
Fig.25c

Fig.25d
Fig.25e

Fig.25f
Fig.25g

Fig.26
Fig.27a

Fig.27b
Fig.27c

Fig.27d
Fig.28a

Fig.28b
Fig.28c

Fig.28d
Fig.28e

Fig.29a
Fig.29b

Fig.29c
Fig.29d

Fig.30a
Fig.30b

Fig.30c
Fig.30d

Fig.30e
Fig.30f

Fig.31a
Fig.31b

Fig.31c
Fig.31d

Fig.31e
Fig.31f

Fig.32a
Fig.32b

Fig.32c
Fig.32d

Fig.33a
Fig.33b

Fig.33c
Fig.34a

Fig.34b
Fig.34c

Fig.34d
Fig.34e

Fig.34f
Fig.35a

Fig.35b
Fig.35c

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

Fig.21 Fig.22a Fig.22b Fig.22c Fig.23a Fig.23b Fig.23c Fig.23d Fig.23e Fig.23f Fig.24a Fig.24b Fig.24c Fig.24d Fig.24e Fig.25a Fig.25b Fig.25c Fig.25d Fig.25e Fig.25f Fig.25g Fig.26 Fig.27a Fig.27b Fig.27c Fig.27d Fig.28a Fig.28b Fig.28c Fig.28d Fig.28e Fig.29a Fig.29b Fig.29c Fig.29d Fig.30a Fig.30b Fig.30c Fig.30d Fig.30e Fig.30f Fig.31a Fig.31b Fig.31c Fig.31d Fig.31e Fig.31f Fig.32a Fig.32b Fig.32c Fig.32d Fig.33a Fig.33b Fig.33c Fig.34a Fig.34b Fig.34c Fig.34d Fig.34e Fig.35 Fig.36a Fig.36b Fig.36c


     In another method, slightly different shaped heart curves are obtained and shown in Heart curve II of "Heart Curves II".

     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.36.    Herein, is nearly fixed in the range of from 0.95 to 0.98.

In ( , ) plain
Fig.36a   The region of the parameters
in which a heart curve may be found
Herein, pink colored area represents its region, and dark
pink colored dots represents data points which are obtained
when heart curves are displayed as above.   Herein, is
nearly fixed in the range of from 0.95 to 0.98.



In ( , ) plain
Fig.36b   The region of the parameters and
in which a heart curve may be found
Herein, pink colored area represents its region, and dark
pink colored dots represents data points which are obtained
when heart curves are displayed as above.   Herein, is
nearly fixed in the range of from 0.95 to 0.98.

     In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(1), (2), (3a), (3b) and (5), a C++ program is given by C++_program_2b_2.
    By executing the C++ program, a text file named "heart_curve_2b_2.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     The case using the tangential function as


     As the conversion equation , we consider the following equation using a linear combination of the tangential function and the proportional relation , which combination nearly satisfies the characteristics given in Fig.3.
           ,                      (6)
where and indicate the weighting ratios in respect to the tangent function and the proportional relation respectively.      Therefore, is regarded as the coefficient expressing so-called degree of strength of the conversion.      is the arbitrary constant which is introduced to adjust the tangential function to the boundaries of the curve in Fig.3 under the condition of as a principle.

     By calculating Eqs.(1), (2), (3a), (3b) and (6), the coordinate data of heart curves are obtained.    Examples of such obtained curves are shown in Fig.37, where is generally taken because the coefficient relates only to the whole syze of the curve and does not relate to the shape of the curve.

Fig.37a
Fig.37b

Fig.37c
Fig.37d

Fig.37e
Fig.37f

Fig.37g
Fig.37h

Fig.37i
Fig.37j

Fig.37k
Fig.37l

Fig.37m
Fig.37n

Fig.37o
Fig.37p

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

Fig.37a Fig.37b Fig.37c Fig.37d Fig.37e Fig.37f Fig.37g Fig.37h Fig.37i Fig.37j Fig.37k Fig.37l Fig.37m Fig.37n Fig.37o Fig.37p



     In purpose to calculate the numerical coordinates data of some closed curve by Eqs.(1), (2), (3a), (3b) and (6), a C++ program is given by C++_program_2b_3.
     By executing the C++ program, a text file named "heart_curve_2b_3.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 the closed curve with the use of a graph wizard attached on the excel file.


6    The case using the inverse hyperbolic-tangential function as


     As the conversion equation , we consider the following equation using a linear combination of the inverse hyperbolic-tangential function and the proportional relation , which combination nearly satisfies the characteristics given in Fig.3.
           ,                      (7)
where and indicate the weighting ratios in respect to the inverse hyperbolic-tangential function and the proportional relation respectively.    Therefore, is regarded as the coefficient expressing so-called degree of strength of the conversion.     is the arbitrary constant which is introduced to adjust the inverse hyperbolic-tangent function to the boundaries of the curve in Fig.3 under the condition of .

     By calculating Eqs.(1), (2), (3a), (3b) and (7), the coordinate data of heart curves are obtained.    Examples of such obtained curves in the case of are shown in Fig.38, where decides only the size and does not relate to the shape.

Fig.38a
Fig.38b

Fig.38c
Fig.38d

Fig.38e
Fig.38f

Fig.38g
Fig.38h

Fig.38i
Fig.38j

Fig.38k
Fig.38l

Fig.38m
Fig.38n

Fig.38o

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

Fig.38a Fig.38b Fig.38c Fig.38d Fig.38e Fig.38f Fig.38g Fig.38h Fig.38i Fig.38j Fig.38k Fig.38l Fig.38o Fig.38p Fig.38m




     In purpose to calculate the numerical coordinates data of a heart curve by Eqs.(1), (2), (3a), (3b) and (7), a C++ program is given by C++_program_2b_4.
     By executing the C++ program, a text file named "heart_curve_2b_4.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.


7    The case using the inverse Fermi distibution function as


     Fermi distibution function is the energy distribution function of Fermi particles (for example, electrons) in quantum mechanics and is given by the followings;
           ,                      (8)
where denotes the particle energy normalized by the thermal energy.      The inverse function of Eq.(8) gives the curve resembling to that shown in Fig.3.
     Thus, as the conversion equation , we consider the following equation using a linear combination of the inverse Fermi distibution function and the proportional relation , which combination nearly satisfies the characteristics given in Fig.3.
           ,                      (9)
where and indicate the weighting ratios in respect to the inverse Fermi distibution function and the proportional relation respectively.    Therefore, is regarded as the coefficient expressing so-called degree of strength of the conversion.      is the arbitrary constant which is introduced to adjust the inverse Fermi distibution function to the boundaries of the curve in Fig.3 under the condition of .

     By calculating Eqs.(1), (2), (3a), (3b) and (9), the coordinate data of heart curves are obtained.    Examples of such obtained curves in the case of are shown in Fig.39, where decides only the size and does not relate to the shape.

Fig.39a
Fig.39b

Fig.39c
Fig.39d

Fig.39e
Fig.39f

Fig.39g
Fig.39h

Fig.39i
Fig.39j

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

Fig.39a Fig.39b Fig.39c Fig.39d Fig.39e Fig.39f Fig.39g Fig.39h Fig.39i Fig.39j




     In purpose to calculate the numerical coordinates data of a heart curve by Eqs.(1), (2), (3a), (3b) and (9), a C++ program is given by C++_program_2b_5.
     By executing the C++ program, a text file named "heart_curve_2b_5.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.



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Updated: 2011.12.28, edited by N. Yamamoto.
Revised on Jan. 10, 2012, Jan. 21, 2012, Jan. 31, 2012, Aug. 27, 2013, Jan. 10, 2014, Jan. 15, 2014, Mar. 16, 2015, Dec. 20, 2015, Jul. 23, 2016, May 05, 2020, May 11, 2020, May 16, 2020, Jan. 19, 2021, May 08, 2021, Feb. 27, 2022 and Mar. 08, 2022.