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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. 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. ![]() where ![]() ![]() The location of the coordinate origin will be intended to be replaced to the bottom of the Cardioid in Fig.1. In this figure, ![]() ![]() where ![]() ![]() If we substitute Eqs.(1) and (2) into the above equation, we obtain
![]() where ![]() The newly defined phase angle ![]() ![]() 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 ![]() ![]() ![]() ![]() If we substitute Eq.(4) into Eq.(5), the conversion equation from ![]() ![]() ![]() The orthogonal coordinate expression of the heart curve may be written as the following two equations; ![]() ![]() where ![]() ![]() By calculating Eqs.(2), (3), (6), (7) and (8), the ![]() ![]() ![]() |
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Moreover, in order to obtain beautiful shape of heart figure, both of the coefficient ![]() ![]() ![]() ![]() and ![]() where ![]() By calculating Eqs.(9a), (9b), (10a), (10b) and (11), the ![]() ![]() ![]() |
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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 ![]() 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 ![]() ![]() ![]() |
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![]() ![]() ![]() 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. |
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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. ![]() where ![]() ![]() By calculating Eqs.(9), (10a), (11) and (12), the ![]() ![]() |
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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. ![]() where ![]() ![]() By calculating Eqs.(9), (10a), (11) and (13), the ![]() ![]() |
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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. ![]() where ![]() ![]() By calculating Eqs.(2), (3), (6), (7) and (14), the ![]() ![]() |
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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. ![]() where ![]() ![]() By calculating Eqs.(2), (3), (6), (7) and (15), the ![]() ![]() |
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When the above figures are painted, these are shown in the followings.![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
8. Method 7
![]() ![]() ![]() ![]() where ![]() 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 ![]() ![]() Therefore, we must use the following equation, which is led from the equation situated immediately on Eq.(2), instead of Eq.(2). ![]() Moreover, we must use the following equation, which is led from the equation situated immediately on Eq.(3), instead of Eq.(3). ![]() 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 ![]() ![]() |
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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 ![]() By calculating Eqs.(16), (17), (18), (6), (7) and (15), the ![]() ![]() |
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When the above figures are painted, these are shown in the followings.![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
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