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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. 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.
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
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 .
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. 

































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. 






































































































































































When the above figures are painted, these are shown in the followings.
In another method, the better shaped heart curves are obtained and shown in 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. 


Herein, pink colored area represents its region, and blue 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. 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. 












































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








































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














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




















When the above figures are painted, these are shown in the followings.
8. Method 7
, (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. 
















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
















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

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