Heart Curves II

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JAPANESE

Heart Curves II

Nobuo YAMAMOTO

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1.  Preface

     Many types of heart curves are seen in the site of   Heart Curve as one of representation.
     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.

Fig.1 Cardioid

2.   Method 1

     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

Fig.2 The definition of

, (3)
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

Fig.3 Conversion of the phase angle
from

into

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

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.


Fig.4	Fig.5	Fig.6a	Fig.6b =30°, b=20%


Fig.6c =30°, b=23%	Fig.7a =45°, b=25%	Fig.7b =45°, b=30%	Fig.7c =45°, b=35%


Fig.8a =60°, b=35%	Fig.8b =60°, b=40%	Fig.8c =60°, b=45%


Fig.9a =90°, b=50%	Fig.9b	Fig.10 =120°, b=80%

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

In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(2), (3), (6), (7) and (8), a C++ program is given by

.
By executing the C++ program, a text file named "heart_curve_a.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.

3. Method 2
In this section, we try to make a corner on the bottom of Cardioid by a nonlinear conversion of the phase angle and try to bring the Cardioid into the shape of a heart curve. Specifically, we converse the phase angle

of a Cardioid (in Fig.11a) into the phase angle

of a heart curve (in Fig.11b) in the manner as shown in Fig.11c. If the conversion function is given as

, the conversion relation is expressed as

. (9a)
Here, we adopt the following equation using a linear combination of the square root function and the proportional relation

, which satisfies the characteristics given in Fig.11c.

, (9b)
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.11c.
Thus, we can make a corner on the bottom of the Cardioid.
Further, a little different type of heart curves are described in

in the case that Fig.11c is expressed by using inverse-trigonometrical function instead of using the square root function of Eq.(9b).


Fig.11a Cardioid	Fig.11b Heart curve	Fig.11c 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.

, (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.


Fig.12a b=1, c=0.3, d=1	Fig.12b b=1, c=0.4, d=1	Fig.12c b=1, c=0.4, d=1.1	Fig.13a b=1.2, c=0.4, d=1


Fig.13b b=1.2, c=0.5, d=1.1	Fig.13c b=1.2, c=0.6, d=1.2	Fig.14a b=1.5, c=0.5, d=0.95	Fig.14b b=1.5, c=0.5, d=1


Fig.14c b=1.5, c=0.6, d=1	Fig.14d b=1.5, c=0.6, d=1.05	Fig.14e b=1.5, c=0.6, d=1.1	Fig.14f b=1.5, c=0.6, d=1.15


Fig.14g b=1.5, c=0.6, d=1.2	Fig.14h b=1.5, c=0.65, d=1	Fig.14i b=1.5, c=0.65, d=1.1	Fig.14j b=1.5, c=0.65, d=1.2


Fig.14k b=1.5, c=0.65, d=1.3	Fig.14l b=1.5, c=0.7, d=1	Fig.14m b=1.5, c=0.7, d=1.2	Fig.14n b=1.5, c=0.7, d=1.4


Fig.15a b=1.65, c=0.6, d=1	Fig.15b b=1.65, c=0.6, d=1.1	Fig.16a b=1.8, c=0.3, d=0.7	Fig.16b b=1.8, c=0.3, d=0.8


Fig.17a b=1.8, c=0.5, d=0.8	Fig.17b b=1.8, c=0.5, d=0.9	Fig.17c b=1.8, c=0.5, d=1	Fig.18a b=1.8, c=0.6, d=1


Fig.18b b=1.8, c=0.6, d=1.1	Fig.19a b=1.8, c=0.67, d=1	Fig.19b b=1.8, c=0.67, d=1.05	Fig.19c b=1.8, c=0.67, d=1.1


Fig.19d b=1.8, c=0.67, d=1.15	Fig.19e b=1.8, c=0.67, d=1.2	Fig.19f b=1.8, c=0.7, d=1	Fig.19g b=1.8, c=0.7, d=1.05


Fig.19h b=1.8, c=0.7, d=1.1	Fig.19i b=1.8, c=0.7, d=1.15	Fig.19j b=1.8, c=0.7, d=1.2	Fig.19k b=1.8, c=0.7, d=1.25


Fig.19l b=1.8, c=0.7, d=1.3	Fig.19m b=1.8, c=0.7, d=1.35	Fig.19n b=1.8, c=0.7, d=1.4	Fig.20a b=2, c=0.3, d=0.7


Fig.20b b=2, c=0.3, d=0.8	Fig.20c b=2, c=0.5, d=0.7	Fig.20d b=2, c=0.5, d=0.8	Fig.21a b=2, c=0.6, d=0.9


Fig.21b b=2, c=0.65, d=1	Fig.21c b=2, c=0.7, d=1	Fig.21d b=2, c=0.7, d=1.05	Fig.21e b=2, c=0.7, d=1.1


Fig.21f b=2, c=0.7, d=1.15	Fig.21g b=2, c=0.7, d=1.2	Fig.21h b=2, c=0.7, d=1.25	Fig.21i b=2, c=0.7, d=1.3


Fig.21j b=2, c=0.7, d=1.4	Fig.21k b=2, c=0.73, d=1	Fig.21l b=2, c=0.73, d=1.1	Fig.21m b=2, c=0.73, d=1.2


Fig.21n b=2, c=0.73, d=1.3	Fig.21o b=2, c=0.73, d=1.4	Fig.22a b=2.2, c=0.7, d=0.8	Fig.22b b=2.2, c=0.7, d=0.9


Fig.22c b=2.2, c=0.7, d=1	Fig.22d b=2.2, c=0.7, d=1.05	Fig.22e b=2.2, c=0.7, d=1.1	Fig.22f b=2.2, c=0.7, d=1.15


Fig.22g b=2.2, c=0.75, d=1	Fig.22h b=2.2, c=0.75, d=1.1	Fig.22i b=2.2, c=0.75, d=1.2	Fig.22j b=2.2, c=0.75, d=1.25


Fig.22k b=2.2, c=0.75, d=1.3	Fig.23a b=2.5, c=0.7, d=0.8	Fig.23b b=2.5, c=0.78, d=1	Fig.23c b=2.5, c=0.78, d=1.1


Fig.23d b=2.5, c=0.78, d=1.2	Fig.23e b=2.7, c=0.78, d=0.95	Fig.23f b=2.7, c=0.78, d=1	Fig.23g b=2.7, c=0.78, d=1.05


Fig.23h b=2.7, c=0.82, d=1	Fig.23i b=2.7, c=0.82, d=1.1	Fig.23j b=2.7, c=0.82, d=1.2

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.


In ( , ) plain	In ( , ) plain

Fig.24 The region of the parameters , and 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.

In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(9a), (9b), (10a), (10b) and (11), a C++ program is given by

.
By executing the C++ program, a text file named "heart_curve_b.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.25a	Fig.25b	Fig.25c	Fig.25d


Fig.25e	Fig.25f

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.


Fig.26a	Fig.26b	Fig.26c	Fig.26d


Fig.27a	Fig.27b	Fig.27c	Fig.27d


Fig.28a	Fig.28b	Fig.28c	Fig.28d


Fig.29a	Fig.29b	Fig.29c	Fig.29d


Fig.30a	Fig.30b	Fig.31a	Fig.31b


Fig.32a	Fig.32b

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

In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(9), (10a), (11) and (12), a C++ program is given by

.
By executing the C++ program, a text file named "heart_curve_c.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.   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.


Fig.33a	Fig.33b	Fig.34a	Fig.34b


Fig.34c	Fig.34d	Fig.35a	Fig.35b


Fig.35c	Fig.35d	Fig.36a	Fig.36b


Fig.36c	Fig.36d	Fig.37a	Fig.37b


Fig.38a	Fig.38b	Fig.39a	Fig.39b

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

In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.Eqs.(9), (10a), (11) and (13), a C++ program is given by

.
By executing the C++ program, a text file named "heart_curve_d.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.

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.


Fig.40	Fig.41	Fig.42	Fig.43a


Fig.43b	Fig.44	Fig.45

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

In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(2), (3), (6), (7) and (14), a C++ program is given by

.
By executing the C++ program, a text file named "heart_curve_e.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.

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.


Fig.46	Fig.47	Fig.48	Fig.49a


Fig.49b	Fig.50a	Fig.50b	Fig.51a


Fig.51b	Fig.51c

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

In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(2), (3), (6), (7) and (15), a C++ program is given by

.
By executing the C++ program, a text file named "heart_curve_f.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.

8. Method 7

Fig.52 The distorted Cardioid in which the moving
radius is multiplied by .

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).

, (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

.
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.


Fig.53a	Fig.53b	Fig.53c	Fig.53d


Fig.54a	Fig.54b	Fig.54c	Fig.54d

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

In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(16), (17), (18), (6), (7) and (14), a C++ program is given by

.
By executing the C++ program, a text file named "heart_curve_g.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.

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.


Fig.55a	Fig.55b	Fig.55c	Fig.55d


Fig.56a	Fig.56b	Fig.56c	Fig.56d

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

In purpose to calculate the numerical coordinates data of a single heart curve by Eqs.(16), (17), (18), (6), (7) and (15), a C++ program is given by

.
By executing the C++ program, a text file named "heart_curve_h.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.

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updated: 2009.02.16, edited by N. Yamamoto
Revised in Apr. 23, 2010, Dec. 10, 2011 and Aug. 05, 2012.