Using the way described above, Eq.(1) is rewritten as in the following;
.
(6)
From the above equation, we obtain the radius vector r in the expression of a asteroid or a shape of a flower having n number of corners as
.
(7)
For the expression in the orthogonal coordinates, the usual variables conversion is applied as
,
(8)
and
.
(9)
By calculating Eqs.(7), (8) and (9) with the use of computer as giving the value of
in the range of , we obtain the figures as shown below.
(1) In the case of star graphics (n=5)
When the above figures are painted, the following ones are drawn.
(2) In the case that a=0.81, =0.1*,
b=0.14 and c=1
When the above figures are painted, the following ones are drawn.
(3) In the case that a=0.83, =0.5*
and b=c=1
When the above figures are painted, the following ones are drawn.
(4) In the case that a=0.83, =0 and b=c=1
When the above figures are painted, the following ones are drawn.
(5) In the case that n=5 and b=c=1
When the above figures are painted, the following ones are drawn.
(6) In the case that n=5, a=0.7, b=0.2 and c=0.3
When the above figures are painted, the following ones are drawn.
(7) In the case that n=5, a=0.81, b=0.022 and c=1 (There are also the cases of <0.)
When the above figures are painted, the following ones are drawn.
(8) In the case that a=0.81, =0.1* and c=1
When the above figures are painted, the following ones are drawn.
(9) In the case that a=0.81, =0.1* and c=1 (II)
When the above figures are painted, the following ones are drawn.
(10) In the case of the other parameters (There are also the cases of <0.)
When the above figures are painted, the following ones are drawn.
The various types of flower shapes are continued to
.
［Reference］
Formal Procedure
For the change of coordinates from the orthogonal ones (x, y) to the polar ones in Eq.(1),
the usual variables conversion is applied as
,
(10)
and
.
(11)
In order that the display having four corners as shown in Fig.1 is expanded to one having the arbitrary n number of corners, and moreover,
in order that one of the corners is located on the top of the figure, the phase angle
is replaced to .
Thus, the conversion formulas are given by the followings instead of Eqs.(10) and (11).
,
(12)
and
.
(13)
If we substitute Eqs.(12) and (13) into Eq.(1), we obtain
.
(14)
where the constants a, b and c are the positive real numbers, and only the constant a is restricted by the follows;
.
(15)
By calculating Eq.(14) as giving the value of in the range of ,
the radius vector r should be obtained theoretically.
However, this calculation is generally impossible analytically except for a limited value of n.
