HOME  JAPANESE 
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  WolframAlpha and Heart Curve  from Wolfram MathWorld as representations.
In this page, it is tried that a Cardioid is reformed into a heart curve.
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.
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. 






































































































































































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". 




in which a heart curve may be found 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. 












































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. 








































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. 














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




















When the above figures are painted, these are shown in the followings.
8. Method 7
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 5. As the concept in the method 5 is based on the method 1, should be multiplied by . Then, the following equation into which Eq.(1) is rewritten is corresponding to Eqs.(10a) and (10b). 
















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). 
















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

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