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Heart Curve II b

Nobuo YAMAMOTO

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 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 ), 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.

In another method, slightly different shaped heart curves are obtained and shown in 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 .      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.34f Fig.35a

 Fig.35b Fig.35c

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

In another method, slightly different shaped heart curves are obtained and shown in 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 .    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 .      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.

 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 .      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 .      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.