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Heart Curves II b

Nobuo YAMAMOTO

     1.  Preface

     Many types of heart curves are seen in the site of   Heart Curve  as one of representation.
     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.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


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



     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.


     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.35
Fig.36a
Fig.36b
Fig.36c


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



     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.



     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.38o
Fig.38p
Fig.38m


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



     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.



     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
Additional editions were done in Jan. 10, 2012, in Jan. 21, 2012 and in Jan. 31, 2012.