Equation Expressing Star or Flower Graphics

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Equation Expressing Star or Flower Graphics

Nobuo YAMAMOTO

You can copy and use all the figures in this page freely.

    From "Equation of Concave Circle" (refer to ) discribed in another page of this home page, we have tried to derive the equation expressing star or flower graphics.

    The equation of the concave circle is written as
             ,                     (1)
where the constants a, b and c are the positive real numbers, and only the constant a is restricted by the following condition;
             .                                 (2)
    A typical example of the original concave circle calculated by Eq.(1) is shown in Fig.1.       In Eq.(1), the coordinate variables x and y will be replaced into the corresponding variable of the phase angle and the corresponding variable of the radius vector r respectively.     The latter means that the region of y>0 will be treated, in another word, the region of the upper half plane in Fig.1 will be treated.
    In the first, the replacement into the phase angle is done in the manner as shown in Fig.2 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, one of the corners is located on the top of the figure.     Such a replacement may be written as
             ,                     (3)
where is the maximum value of x in the original equation of concave circle, and is written as
             .                                 (4)
    In the next, the replacement into the radius vector r is written as
             .                     (5)


Fig.1 An example of the original concave circle obtained by Eq.(1)	Fig.2 A manner of replacement of x into

    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.

    In purpose to calculate the numerical coordinates data of star or flower graphics, a C++ program originated from Eqs.(7), (8) and (9) is given by .
    By executing the C++ program, a text file named "asteroid.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 asteroid or a shape of a flower with the use of a graph wizard attached on the Excel file.
    A mask for the star praphic is prepared in this button or in this button , which are edited from the above obtained curves.
    A mask for flower graphic is prepared in this button , which is edited also from the above obtained curve.
    Please use this mask freely.

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.

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Updated: 2009.01.24, edited by N. Yamamoto.
Revised on Jul.28, 2013, Sep. 01, 2013, Dec. 29, 2013, Dec. 31, 2013, Jan. 13, 2014, Mar. 16, 2015, May 05, 2020, Jan. 20, 2021, May 15, 2021, Mar. 07, 2022 and Mar. 08, 2022.