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1. Preface
We try to produce shape of flower by calculation with the use of computer as an application of "heart curves II" treated in the other page of this home page. According to two kinds of methods written in the page of "heart curves II", these two methods will be followed also in this page. 2. Method 1
Method 1 is to make a heart curve from a Cardioid by forming an angle on its bottom simply, where the heart curve will be used as a petal of a flower, 2.1 In the case of no change in the length and width of heart curve to be used as a petal of flower
The original Cardioid is shown in Fig,1a, and the heart curve, which is produced from this original Cardioid by the method 1 as described in the page of "heart curves II", is shown in Fig.1b. We intend to form a shape of flower by arranging heart curves around a center circle as petals of a flower.
, (1)
















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Added in Sep 21, 2012.  
［2.1 Additional II］ In the case that each petal contacts with the center circle with an arbitral width of phase angle of its circle
The base of petals may actually be connected to the flower floor with a narrow width. In the simulation of this page, this fact is corresponding to that the petals are connected to a center circle with a narrow width. 
 
Added in Sep. 30, 2012.  
2.2 In the case of change in the length and width of heart curve to be used as a petal of flower
In this case, we return to a heart curve (as shown in Fig.1 and which is used as the material of a petal) in the (x, y) coordinate expression. This expression is described in the following two equations with the ratio of expansion and contraction in the y direction. 
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［2.2 Additional I］
In the case of overlap among the petals of a flower

When the first half part of the above figures is painted, the following figures are drawn.
 
Added in Sep 21, 2012.  
［2.2 Additional II］
In the case that each petal contacts with the center circle with an arbitral width of phase angle of its circle 
 
c=0.3, l=0  c=0.3, l=0.5  c=0.3, l=1  c=0.3, l=2 
 
Added in Oct. 01, 2012.  
3. Method 2
Method 2 is to make a heart curve from a Cardioid by forming an angle on its bottom in application of some conversion equation of the phase angle, where the heart curve will be used as a petal of a flower, 
from into 
The equations obtained in the page "heart curves II" and used also in this page are the following three ones;
The origin point in Fig.31b is needed to be moved to the bottom of the heart curve. Although Eq.(16) is not varied in such a displacement, Eq.(17) should be varied in the following equation.
Newly, the expression of the whole shape of a flower in the orthogonal coordinates are given as 3.1.1 In the case using Fig.33 As seen in Fig.33, the minimum and maximum phase angles of the heart curve possess the proper meanings. The values of these angles will be obtained by software of C program. On this base, the (x, y) coordinate data of shape of flower are calculated by Eqs.(15) to (24). Examples of such calculated shape of flower in the case of are shown below. 
e=0.05 
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［3.1.1 Additional I］ In the case of overlap among the petals of a flower In order to make overlap among the petals of a flower, it is prefered that the width of a single petal is intended to be increased. If the inclination of the lines in Fig.32 is increased, the width of a single petal is increased. Therefor, as the inclimentation factor of this inclination, the following equation is used instead of Eq.(22). . (C1) Then, the (x, y) coordinate data of the shape of a flower are calculated by Eqs.(15) to (21), Eqs.(23), (24) and Eq.(C1), Examples of such calculated shape of flower in the case of are shown below. 
 
d=1, e=0.5, k=1, p=0.5  d=1, e=0.5, k=1, p=1  d=1, e=0.5, k=1, p=1.5  d=1, e=0.5, k=1, p=2 
d=1, e=0.5, k=1, p=3  d=1, e=0.7, k=1, p=1.2 
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Added in Sep 22, 2012.  
［3.1.1 Additional II］ In the case that each petal contacts with the center circle with an arbitral width of phase angle of its circle
We can describe the solution of this problem as the followings likely as in [2.1 Additional II] of the section 2.1 and likely as in [2.2 Additional II] of the foregoing section. 
 
d=1, e=0.4, k=1, l=0  d=1, e=0.4, k=1, l=0.5  d=1, e=0.4, k=1, l=1  d=1, e=0.4, k=1, l=2.5 
 
Added in Oct. 01, 2012.  
3.1.2 In the case using Fig.34
As seen in Fig.34, as the minimum and maximum phase angles are defined at the bottom of the heart curve, the values of these angles are generally different from ones described in the section 3.1.1. These angles are obtained analytically as 
 
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［3.1.2 Additional I］ In the case of overlap among the petals of a flower
We can use all the equations of Eqs.(15) to (21), Eqs.(23), (24) and Eq.(C1) as likely as used in the 3.1.1 Additional, 
 
d=1, e=0.5, k=1, p=0.5  d=1, e=0.5, k=1, p=1  d=1, e=0.5, k=1, p=1.5  d=1, e=0.5, k=1, p=2 
When the above figures are painted, the following ones are drawn.
 
Added in Sep 22, 2012.  
［3.1.2 Additional II］ In the case that each petal contacts with the center circle with an arbitral width of phase angle of its circle
We can describe the solution of this problem as the followings likely as in [2.1 Additional II] of the section 2.1, likely as in [2.2 Additional II] of the the section 2.2 and likely as in [3.1.1 Additional II] of the foregoing section. 
 
d=1, e=0.4, k=1, l=0  d=1, e=0.4, k=1, l=0.5  d=1, e=0.4, k=1, l=1  d=1, e=0.4, k=1, l=2.5 
 
Added in Oct. 01, 2012.  
3.2 In the case of using heart curves as petals of a flower with simplification
The process treating in the previous section 3.1 is a little complicated for holding a correct shape of heart curve. So, we divert the phase angle , which is given by Eq.(12) as an angle before conversion, instead of the correct phase angle , which is given by Eq.(22) and is shown in Fig.32, as a phase angle of shape of a flower. The conversion of the phase angle from (a single heart curve expressed in Fig.31b) into (the shape of a flower) is like that , and are replaced to , and respectively in Eq.(22). 
 
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［3.2 Additional I］ In the case of overlap among the petals of a flower We can also use all the equations of From Eqs.(15) to (21) and Eq.(C1) as likely as used in the 3.1.1 Additional, Moreover, we use Eqs.(27) and (28) described just above. Then, the (x, y) coordinate data of the shape of a flower are calculated by Eqs.(15) to (21), Eq.(C1), Eqs.(27) and (28), Examples of such calculated shape of flower in the case of are shown below. 
 
c=0.1, d=1, e=1, p=0.5  c=0.1, d=1, e=1, p=1  c=0.1, d=1, e=1, p=1.5  c=0.1, d=1, e=1, p=2  c=0.1, d=1, e=1, p=3 
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Addid in Sep 22, 2012.  
［3.2 Additional II］ In the case that each petal contacts with the center circle with an arbitral width of phase angle of its circle
We can describe the solution of this problem as the followings likely as in [2.1 Additional II] of the section 2.1, likely as in [2.2 Additional II] of the the section 2.2 and likely as in [3.1.1 Additional II] of the foregoing section. 
 
c=0.5, d=1, e=0.4, l=0  c=0.5, d=1, e=0.4, l=0.5  c=0.5, d=1, e=0.4, l=1  c=0.5, d=1, e=0.4, l=2.5 
 
Added in Oct. 02, 2012.  
4. In the case of using Cardioid directly as a petal of flower
In this section, we treat the case of using Cardioid directly as a petal of flower. 4.1 In the case of no overlap among the petals of a flower
The conversion of the phase angle from (that of a single Cardioid) to (that of the shape of a flower) is like that , and are replaced to , t and respectively in Eq.(22).


 
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4.2 In the case of overlap among the petals of a flower In this case, the conversion of the phase angle from (that of a Cardioid) to (that of the shape of a flower) is like that and are replaced to and respectively in Eq.(34). , (351) where indicates the th number's petal of a flower. Likely as in the previous section 4.1, as the movement of the origin point does not make influence to the conversion of the phase angle for simplification, the each petal is not taken over the original Cardioid. Further, as an index of the overlap, the evolved equation of Eq.(351) is shown in the following. , (352) where indicates the th number's petal of a flower. Then, for example, the index is specified as in the case of the previous section 4.1 and specified as in the case that adjacent petals are fully overlapped. Calculating Eq.(18), Eqs.(29) to (33) and either Eq.(351) or Eq.(351), examples of shape of flower in the case of are obtained as shown below. 
 
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edited in Mar. 15, 2009 & reviced in Sep. 11, 2012  
4.3 In the case that each petal contacts with the center circle with an arbitral width of phase angle of its circle
We can also use Eq.(18) and Eqs.(29) to (33). 
 
 
Added in Oct. 02, 2012.  
5. In the case of using a circle directly as a petal of flower
5.1 The first deformation of a circle used as a petal
We assume a deformed circle given by the followings in the orthogonal coordinates; 
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5.1.2 In the case of overlap among the petals of a flower
In this case, Eq.(351) or (352) can be used as the equation of transformation from (the phase angle of a circle) to (the phase angle around the shape of a flower), where has to be replaced to .

d=2, e=0, p=2  d=2, e=1, p=2  d=3, e=1, p=2  d=3, e=1, p=2 
d=2, e=2, p=0.5  d=2, e=2, p=1  d=2, e=2, p=1.5  d=2, e=2, p=2 
d=2, e=2, p=3  d=2, e=2, p=4 
When the above figures of Figs.85 to 83b and Fig.96f are painted, they are drawn as in the folloings. If the each of four figures in the left side is clicked, an expanded line drawing appears.
 
edited in Mar. 15, 2009 & reviced in Sep. 14, 2012  
5.1.3 In the case that each petal contacts with the center circle with an arbitral width of phase angle of its circle
We can also use Eqs.(27), (31), (36) and (37). 
 
d=3, e=2, l=0  d=3, e=2, l=0.5  d=3, e=2, l=1  d=3, e=2, l=2.5 
d=0.85, e=0.3, l=0.4  d=0.8, e=0.35, l=1.5  d=1, e=0.37, l=1.5 
When the above figures of Fig.F1f and Fig.F1g are painted, they are drawn as in the folloings. If clicked, an expanded line drawing appears.
 
Added in Oct. 02, 2012.  
5.1.4 The cases that the principal terms in Eqs.(36) and (37) are exchaneged mutually and/or the signs in Eqs.(36) and (37) are chaneged variously
We treat several cases that the principal terms in Eqs.(36) and (37) are exchanged mutually and /or the signs contained in Eqs.(36) and (37) are chaneged variously while process of calculations is obeyed to that described in the section 5.1.3. 5.1.4 (i) The case 1
The following two equations are proposed for replacing Eqs.(36) and (37). 


d=1, e=1, l=0  d=1, e=1, l=0.5  d=1, e=1, l=1  d=1, e=1, l=1.5 
 
Added in Oct. 16, 2012.  
5.1.4 (ii) The case 2 The following two equations are proposed for replacing Eqs.(36) and (37). . (E3) , (E4) where the last term of Eq.(E4) is intended to move the origin to the bottom of the deformed circle. If the (x, y) coordinate data of the shape of a flower are calculated by Eqs.(E3), (E4), (31), (27), (D3) and (D4), some examples of figures in the case of are obtained as shown below. 
 
d=1, e=1, l=0  d=1, e=1, l=0.5  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.1.4 (iii) The case 3
The following two equations are proposed for replacing Eqs.(36) and (37). 
 
d=1, e=1, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.1.4 (iv) The case 4
The following two equations are proposed for replacing Eqs.(36) and (37).

 
d=1, e=1, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.1.4 (v) The case 5

 
d=1, e=1, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.1.4 (vi) The case 6

 
d=1, e=1, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.1.4 (vii) The case 7

d=1, e=1, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.2 The second deformation of a circle used as a petal
Calculating Eqs.(31) to (34), Eqs.(38) and (39), examples of shape of flower in the case of are obtained as shown below. 
 
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A part of the figures and the picture are added in May 14, 2013.  
5.2.2 In the case of overlap among the petals of a flower As likely as in the previous section 5.1, Eq.(351) or (352) can be used as the equation of transformation from (the phase angle of a circle) to (the phase angle around the shape of a flower), where has to be replaced to . Calculating Eqs.(31) to (33), either Eq.(351) or (352), Eqs.(38) and (39), examples of shape of flower in the case of are obtained as shown below. 
d=3, e=1, p=2  d=3, e=1, p2  d=3, e=1, p=2  d=3, e=1, p=2 
d=3, e=3, p=0.5  d=3, e=3, p=1  d=3, e=3, p=1.5  d=3, e=3, p=2 
d=3, e=3, p=3  d=3, e=3, p=4 
When the above figures Figs.89 to 91b and Fig.92f are painted, they are drawn as in the folloings. If the each of four figures in the middle is clicked, an expanded line drawing appears.
 
edited in Mar. 15, 2009 & reviced in Sep. 15, 2012  
5.2.3 In the case that each petal contacts with the center circle with an arbitral width of phase angle of its circle
We can also use Eqs.(27), (31), (38) and (39). 
d=3, e=2, l=0  d=3, e=2, l=0.5  d=3, e=2, l=1  d=3, e=2, l=2.5 
 
Added in Oct. 02, 2012.  
5.2.4 The cases that the principal terms in Eqs.(38) and (39) are exchaneged mutually and/or the signs in Eqs.(38) and (39) are chaneged variously We treat several cases that the principal terms in Eqs.(38) and (39) are exchanged mutually and /or the signs contained in Eqs.(38) and (39) are chaneged variously while process of calculations is obeyed to that described in the section 5.2.3. 5.2.4 (i) The case 1
The following two equations are proposed for replacing Eqs.(38) and (39). 
d=3, e=2, l=1.5  d=2, e=1.5, l=1.5  d=3, e=2, l=1 
When the above figure >Fig.F2f is painted, it is drawn as shown in the folloing. If clicked, an expanded line drawing appears.
 
Added in Oct. 09, 2012.  
5.2.4 (ii) The case 2

d=1, e=1, l=0  d=2, e=1.5, l=0  d=2, e=1.5, l=1.5 
 
Added in Oct. 09, 2012.  
5.2.4 (iii) The case 3 The following two equations are proposed for replacing Eqs.(38) and (39). . (F5) , (F6) where the last term of Eq.(F6) is intended to move the origin to the bottom of the deformed circle. If the (x, y) coordinate data of the shape of a flower are calculated by Eqs.(F5), (F6), (31), (27), (D3) and (D4), some examples of figures in the case of are obtained as shown below. 
d=1, e=1, l=0  d=3, e=2, l=0  d=3, e=2, l=1 
 
Added in Oct. 09, 2012.  
5.2.4 (iv) The case 4
The following two equations are proposed for replacing Eqs.(38) and (39). 
 
d=1, e=1, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.2.4 (v) The case 5

d=1, e=1, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.2.4 (vi) The case 6 The following two equations are proposed for replacing Eqs.(38) and (39). . (F11) , (F12) where the last term of Eq.(F12) is intended to move the origin to the bottom of the deformed circle. If the (x, y) coordinate data of the shape of a flower are calculated by Eqs.(F11), (F12), (31), (27), (D3) and (D4), some examples of figures in the case of are obtained as shown below. 
 
d=1, e=1, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.2.4 (vii) The case 7
The following two equations are proposed for replacing Eqs.(38) and (39). 
d=1, e=1, l=0  d=1, e=1, l=0.5  d=1, e=1, l=1 
 
added in Oct. 16, 2012.  
5.3 The third deformation of a circle used as a petal
We assume a deformed circle given in the followings in the orthogonal coordinates;

 
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5.3.2 In the case of overlap among the petals of a flower As likely as in the previous sections 5.1 and 5.2, either Eq.(351) or (352) can be used as the equation of transformation from (the phase angle of a circle) to (the phase angle around the shape of a flower), where has to be replaced to . Calculating Eqs.(31) to (33), either Eq.(351) or (352), Eqs.(40) and (41), examples of shape of flower in the case of are obtained as shown below. 
 
c=4, d=3, e=1, p=2  c=4, d=3, e=1, p=2  c=4, d=3, e=3, p=0.5  c=4, d=3, e=3, p=1 
c=4, d=3, e=3, p=1.5  c=4, d=3, e=3, p=2  c=4, d=3, e=3, p=3  c=4, d=3, e=3, p=4 
When the above figures Figs.96a and b are painted, they are drawn as in the folloings. If the each of four figures in the left side is clicked, an expanded line drawing appears.
 
edited in Mar. 15, 2009 & reviced in Sep. 16, 2012  
5.3.3 In the case that each petal contacts with the center circle with an arbitral width of phase angle of its circle We can also use Eqs.(27), (31), (40) and (41). Moreover, we can consider the solution of this problem as followgs likely as in [3.2 Additional II] of the section 3.2. So, we also use Eqs.(D3) and (D4). In calculating C program, we take the correction factor as an input variable, and then, we give the arbitral width of phase angle with which each petal contacts to the center circle. The (x, y) coordinate data of the shape of a flower are calculated by Eqs.(27), (31), (40), (41), Eqs.(D3) and (D4), Examples of such calculated shape of flower in the case of are shown below. 
 
d=3, e=2, l=0  d=3, e=2, l=0.5  d=3, e=2, l=1  d=3, e=2, l=2.5 
 
Added in Oct. 02, 2012.  
5.3.4 The cases that the principal terms in Eqs.(40) and (41) are exchaneged mutually and/or the signs in Eqs.(40) and (41) are chaneged variously
We treat several cases that the principal terms in Eqs.(40) and (41) are exchanged mutually and /or the signs contained in Eqs.(40) and (41) are chaneged variously while process of calculations is obeyed to that described in the section 5.3.3. 
d=1, e=1, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.3.4 (ii) The case 2

 
d=3, e=2, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.3.4 (iii) The case 3
The following two equations are proposed for replacing Eqs.(40) and (41). 
d=3, e=2, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.3.4 (iv) The case 4

 
d=3, e=2, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.3.4 (v) The case 5
The following two equations are proposed for replacing Eqs.(40) and (41). 
 
d=3, e=2, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.3.4 (vi) The case 6
The following two equations are proposed for replacing Eqs.(40) and (41). 
d=3, e=2, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
5.3.4 (vii) The case 7
The following two equations are proposed for replacing Eqs.(40) and (41). 
 
d=3, e=2, l=0  d=1, e=1, l=1 
 
Added in Oct. 16, 2012.  
6. Method to draw to set independently both of the phase angles of each petal and a center circle underlying each other
In the previous sections, a flower has been configured with the common phase angle of a center circle and its underlying each petal. In this manner, the width of the petal cannot be extended beyond the area of a predetermined phase angle of the center circle underlying.
The petal as shown in Fig.98b is created by sharpened the bottom of the circle as shown in Fig.98a. The equation of the circle in Fig.98a is described as
In the next, in order that the bottom of the circle is reformed into a petal curve with a corner having the desired angle (as seen in Fig.98b), we converse the phase angle of the circle into the newly defined phase angle of the petal curve linearly as shown in Fig.99. 6.1.1 In the same case as in the previous sections in which common phase angle is applied to both of the each petal and the center circle underlying (reference for comparison)
In this case, the entire shape of a flower is expressed in the following two equations which are the same to Eqs.(6) and (7) in the section 2.1.
Calculating Eq.(42), either Eq.(48) or Eq.(48b), Eqs.(49), (50) and (51), an example of shape of flower in the case of is obtained as shown in Fig.100.
6.1.2 In the case that the phase angle of a center circle is set independently on the phase angle of each petal
According to how each petal contacts with the center circle, we will compose a flower graphics as divided into following three cases. Calculating Eqs.(42), (48) or (48b), (49), (52) and (53), an example of shape of flower in the case of is obtained as shown in Fig.101.
6.1.2 (ii) In the case that each petal contacts with the center circle with an arbitral width of phase angle of its circle
If we take as an arbitral width of phase angle with which each petal contacts with the center circle, the phase angle of the center circle is discribed as , and then, it depends on . 
6.1.2 (iii) In the case to sharpen the tip of the petal treated in the subsection 6.1.2 (ii)

(If clicked, an expanded line drawing appears.)  
(If clicked, an expanded line drawing appears.)  
 
July 09, 2012 & Sep 09, 2012  
6.1.2 (iv) In the case to sharpen the tip of the petal treated in the subsection 6.1.2 (ii)...(part 2)

(If clicked, an expanded line drawing appears.)  
 
 
Sep 09, 2012  
6.2 The case to configure the petals from cardioid There are two methods of the method 1 and 2 described in Section 2 and 3 respectively to convert the underlying heart curve in the form of a petal from cardioid. We will design a flower graphics according to the each method in case by case as below. 6.2.1 In the case with the use of the method 1 described in Section 2
We will treat a flower graphics along to "Method 1" described in Section 2. Unlike Section 2, we set independently of each other between the phase angle of petal and that of a center circle underlying. Therefore, though the following description may overlap with Section 2, its description is intended to be rerecorded for clarity.
. (60) If we substitute Eq.(59) into Eq.(60), the conversion equation from to is obtained as . (61) The orthogonal coordinate expression of the heart curve may be written as the following two equations; . (62) . (63) In order to broaden the variety in the form of a petal, we introduce the ratio of expansion in the y direction, and we rewrite Eq.(63) into the following equation with the use of . . (63b) In the next, we express the heart curve in the polar coordinates again. In this case, the moving radius and the phase angle are given by the followings; , (64) , (65) We will arrange such the obtained heart curves as petals of a flower around a center circle of radius . If numbers of petals are arranged, we assume that the reference angle of the bottom of each petal is taken as the angle going around a flower divided by as given in the following equation; . (66) If the number is assigned to the each of petals, the phase angle of th number's of petal is expressed in the following equation into which the angle described in Eq.(65) is added to . . (67)
On theother hand, if we take as an arbitral width of phase angle with which each petal contacts with the center circle, the phase angle of the center circle in contact with the th petal is described as , and then, it depends on . 
 
 
July 14, 2012  
6.2.2 In the case with the use of the method 2 described in the section 3
We will treat a flower graphics along to "Method 2" described in Section 3. Unlike Section 3, we set independently of each other between the phase angle of petal and that of a center circle underlying. Therefore, though the following discription may overlap with Section 3, its discription is intended to be rerecorded for clarity. 
from into 
Thus, the phase angle is converted as that the bottom of the cardioid becomes to sharpen.
On theother hand, if we take as an arbitral width of phase angle with which each petal contacts with the center circle, the phase angle of the center circle in contact with the th petal is depends on as likely as in the subsection 6.2.1. 
 
e=0, k=1, l=0  e=1, k=1, l=0  e=1, k=1, l=0.5  e=1, k=1, l=1 
e=1, k=1, l=1.5  e=1, k=1.5, l=0.5  e=1, k=2, l=0.5  e=1, k=2, l=0 
c=0, d=1, e=1, k=2, l=1  e=1, k=2, l=0.5  e=0.5, k=1, l=0.3  e=0.5, k=0.7, l=1 
d=1, e=0.2, k=0.8, l=1  d=1, e=0.2, k=0.8, l=1  d=1, e=0.2, k=0.8, l=1  d=1, e=0.2, k=1.5, l=1 
d=1.5, e=0.1, k=1.5, l=1  d=1.5, e=0.2, k=1.8, l=1  d=1.5, e=0.2, k=0.6, l=1 
When Figs.127, 128 and 132d are painted, the following figures are drawn. If clicked, an expanded line drawing appears.
 
July 18, 2012 and May 14, 2013.  
6.2.3 Deformation of the previous subsection 6.2.2

d=1.25, e=0.2, k=0.75, l=1  d=1.3, e=0.2, k=0.75, l=1  d=3, e=2.3, k=1.2, l=0.8 
When the above figures are painted, the following ones are drawn. If clicked, a line drawing appears.
 
July 18, 2012 and May 09, 2013.  
6.2.3 (ii) Deformation II of the previous section 6.2.2
The following equations are replaced to Eqs.(72a) and (72b) respectively in the previous section 6.2.2.

 
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1  d=1.3, e=0.3, k=0.8, l=1  d=3, e=2.8, k=1.4, l=0.7 
When Figs.135c and 136 are painted, the following figures are drawn. If clicked, an expanded line drawing appears.
 
July 22, 2012 and May 09, 2013.  
6.2.3 (iii) Deformation III of the previous section 6.2.2
The following equations are replaced to Eqs.(72a) and (72b) respectively in the previous section 6.2.2. 
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1 
6.2.3 (iv) Deformation IV of the previous section 6.2.2 The following equations are replaced to Eqs.(72a) and (72b) respectively in the previous section 6.2.2. . (84a) . (84b) Calculating Eq.(71), Eqs.(73) to (80), Eqs.(84a) and (84b), examples of shape of flower in the case of are obtained as shown below. 
(If clicked, an expanded line drawing appears.)  
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1 
6.2.3 (v) Deformation V of the previous section 6.2.2 The following equations are replaced to Eqs.(72a) and (72b) respectively in the previous section 6.2.2. . (85a) . (85b) Calculating Eq.(71), Eqs.(73) to (80), Eqs.(85a) and (85b), examples of shape of flower in the case of are obtained as shown below. 
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1  d=1.2, e=0.9, k=0.9, l=0  d=1.2, e=0.6, k=0.9, l=0 
When the above figures Figs.142a and b are painted, they are drawn as in the folloings. If clicked, an expanded line drawing appears.
6.2.3 (vi) Deformation VI of the previous section 6.2.2 The following equation is replaced to Eq.(72a) in the previous section 6.2.2. . (86) Calculating Eq.(71), Eqs.(72b) to (80) and Eq.(86), examples of shape of flower in the case of are obtained as shown below. 
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1 
6.2.3 (vii) Deformation VII of the previous section 6.2.2
The following equation is replaced to Eq.(72b) in the previous section 6.2.2.

 
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1 
6.2.3 (viii) Deformation VIII of the previous section 6.2.2 The following equation is replaced to Eq.(72b) in the previous section 6.2.2.
. (88)

(If clicked, an expanded line drawing appears.)  
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1  d=1.1, e=0.3, k=1.4, l=1 
 
July 22, 2012  
6.2.3 (iX) Deformation IX of the previous section 6.2.2
The following equations are replaced to Eqs.(72a) and (72b) in the previous section 6.2.2.

(If clicked, an expanded line drawing appears.)  
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1  d=1.25, e=0.3, k=0.7, l=1 
6.2.3 (x) Deformation X of the previous section 6.2.2
The following equations are replaced to Eqs.(72a) and (72b) in the previous section 6.2.2.

d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1 
6.2.3 (xi) Deformation XI of the previous section 6.2.2 The following equations are replaced to Eqs.(72a) and (72b) in the previous section 6.2.2. . (91) Calculating Eq.(71), Eqs.(73) to (80) and Eq.(91), examples of shape of flower in the case of are obtained as shown below. 
(If clicked, an expanded line drawing appears.)  
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1 
6.2.3 (xii) Deformation XII of the previous section 6.2.2 The following equations are replaced to Eqs.(72a) and (72b) in the previous section 6.2.2. . (92) Calculating Eq.(71), Eqs.(73) to (80) and Eq.(92), examples of shape of flower in the case of are obtained as shown below. 
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1 
6.2.3 (xiii) Deformation XIII of the previous section 6.2.2
The following equations are replaced to Eqs.(72a) and (72b) in the previous section 6.2.2. 
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1 
6.2.3 (xiv) Deformation XIV of the previous section 6.2.2 The following equations are replaced to Eqs.(72a) and (72b) in the previous section 6.2.2. . (94) Calculating Eq.(71), Eqs.(73) to (80) and Eq.(94), examples of shape of flower in the case of are obtained as shown below. 
d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1 
6.2.3 (xv) Deformation XV of the previous section 6.2.2
The following equations are replaced to Eqs.(72a) and (72b) in the previous section 6.2.2.

d=1, e=0.3, k=0.7, l=1  d=1, e=0.3, k=1, l=1 
 
July 25, 2012  
6.2.4 On the other deformation of the previous section 6.2.2
It may be considered that "the other function" is replaced into the function which is used in the previous subsections of 6.2.3 (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (xi), (xii), (xiv) and (xv) in the section 6.2.3. By the way, as the function is not used in the remained subsections in each of which only a replacement of the sign of positive and negative is done in the equation expressing a petal, so, the remained subsections is not related to the present arguement. 6.2.4.a In the case of replacement of into
Now, we suppose that the function is replaced into the function . Then, the description " *cos(f)*cos(f) " is needed only to be replaced into the description " *fabs(cos(f)) " in the C++ calculation program listed in the each of the related subsections in which the function is used. 
 
July 29, 2012 and Apr. 04, 2013 


May 15, 2013  
6.2.4.b In the case of replacement of into We suppose that the function is replaced into the function . Then, the description " *fabs(sin(f)) " is needed to be replaced into the description " *fabs(cos(f)) " in the C++ calculation program listed in the each of the related subsections in which the function is used. Furthermore, the equation in the C++ calculation program listed in the each of the related subsections 6.2.3 (ii), (iii), (iv), (v), (vii), (viii), (xi) and (xii) of the section 6.2.3, which equation is corresponding to Eq.(80), has to be changed as in the followings. The reason why the each equation must be changed as in the followings is because the radius of the center circle varies affected by such the replacement. On the other hand, there is no change in the each subsection of 6.2.3 (i), (vi), (xiv) and (xv).
" y=d*r*(1c*sin(f)*fabs(sin(f)))*sin(f)+2*a*d*(1+c); " to be replaced into " y=d*r*(1c*sin(f)*fabs(sin(f)))*sin(f)+2*a*d; " in the C++ program of 6.2.3 (ii),
The results of the simulation by C++ program are obtained as shown in Figs.165(i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (xi), (xii), (xiv) and (xv) according to the each method written in the subsections 6.2.3 (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (xi), (xii), (xiv) and (xv) respectively. 
 
July 31, 2012 and Apr. 02, 2013 
 
 
Aug. 01, 2012 and Apr. 02, 2013 
 
Aug. 02, 2012 


May 15, 2013  
6.2.4.c Possibility of replacement of the other function into It is considered that the function such as , and etc. may also be replaced into the function .  
July 31, 2012  
We find that the the simulation of the flower as described in the title is obtained by plotting as in the orthogonal coordinates in the use of as an intermediate variable. An example is shown in Fig.168 where .
number of the petals described above are placed around the center circle of radius . The phase angle of such composed flower can be treated in the same manners as in [2.1 Additional II] and [2.2 Additional II] of Sec. 2, in [3.1.1 Additional II], [3.1.2 Additional II] and [3.2 Additional II] of Sec. 3 and in Sec. 6.1.2(ii).
. (96h) 
(If clicked, an expanded line drawing appears.)  
d=8, e=0, k=1, l=0.2 
(If clicked, an expanded line drawing appears.)  
d=8, e=0, k=1, l=0.2 
 
June 17, 2013  
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