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1. Preface
Many types of heart curves are seen on the sites of Heart Curve  Mathematische Basteleien, First heart Curve  WolframAlpha 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 inversesinusoidal 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 inverseconversion function is given as , the following relation is expressed as . (2) Thus, we should make a corner on the bottom of the Cardioid. 





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 inversesinusoidal function as We adopt the following equation using a linear combination of the inversesinusoidal 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. 




































































































































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, pink colored area represents its region, and dark pink colored dots represents data points which are obtained when heart curves are displayed as above. 
The other types of curves besides the heart shaped ones are also obtained as follows. 










When the above figures are painted, these are shown in the followings.
4. The case using the inversesinusoidal 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 inversesinusoidal 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. 
































































































































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. 


 
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. 


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. 
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 socalled 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. 
































When the above figures are painted, these are shown in the followings.
6 The case using the inverse hyperbolictangential function as As the conversion equation , we consider the following equation using a linear combination of the inverse hyperbolictangential 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 hyperbolictangential function and the proportional relation respectively. Therefore, is regarded as the coefficient expressing socalled degree of strength of the conversion. is the arbitrary constant which is introduced to adjust the inverse hyperbolictangent 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. 






























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




















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

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