Folding Stars using Equations (用方程式摺星星)

Example 1.

(n = 5, ratio = 1)

gnuplot script 1:

n = 5
ratio = 1.0
A = 2*pi/n
c = 2*cos(A)
d = ratio*c*(1 - cos(A))/(1 + cos(A))
max(x,y) = x < y ? y : x
f1(x,y) = max(max(abs(y - c) - d, y*cos(A) - x*sin(A) - c/cos(A)), \
   max(d - abs(y*cos(A) - x*sin(A) - c), y*cos(3*A) - x*sin(3*A) - c + d))
f2(x,y) = f1(y*sin(A) + x*cos(A), y*cos(A) - x*sin(A))
f3(x,y) = f1(y*sin(2*A) + x*cos(2*A), y*cos(2*A) - x*sin(2*A))
f4(x,y) = f1(y*sin(3*A) + x*cos(3*A), y*cos(3*A) - x*sin(3*A))
f5(x,y) = f1(y*sin(4*A) + x*cos(4*A), y*cos(4*A) - x*sin(4*A))
set xrange [-2.5:2.5]
set yrange [-2.5:2.5]
set size ratio -1
set samples 256
set isosamples 256
set contour base
set cntrparam levels discrete 0.0
unset key
unset surface
set table 'f1.dat'
splot f1(x,y)
unset table
set table 'f2.dat'
splot f2(x,y)
unset table
set table 'f3.dat'
splot f3(x,y)
unset table
set table 'f4.dat'
splot f4(x,y)
unset table
set table 'f5.dat'
splot f5(x,y)
unset table
plot 'f1.dat' w l lw 2 lc rgb "#0099ff", \
       'f2.dat' w l lw 2 lc rgb "#0099ff", \
       'f3.dat' w l lw 2 lc rgb "#0099ff", \
       'f4.dat' w l lw 2 lc rgb "#0099ff", \
       'f5.dat' w l lw 2 lc rgb "#0099ff"

Example 2.

(n = 6, ratio = 1)

gnuplot script 2:

n = 6
ratio = 1.0

A = 2*pi/n

c = 1

d = ratio*c/3

max(x,y) = x < y ? y : x

f1(x,y) = max(max(abs(y - c) - d, y - sqrt(3)*x - 4*c), \

   max(d - abs(y*cos(A) - x*sin(A) - c), y*cos(4*A) - x*sin(4*A) - c + d))

f2(x,y) = f1(y*sin(A) + x*cos(A), y*cos(A) - x*sin(A))

f3(x,y) = f1(y*sin(2*A) + x*cos(2*A), y*cos(2*A) - x*sin(2*A))

f4(x,y) = f1(y*sin(3*A) + x*cos(3*A), y*cos(3*A) - x*sin(3*A))

f5(x,y) = f1(y*sin(4*A) + x*cos(4*A), y*cos(4*A) - x*sin(4*A))

f6(x,y) = f1(y*sin(5*A) + x*cos(5*A), y*cos(5*A) - x*sin(5*A))

set xrange [-2.5:2.5]

set yrange [-2.5:2.5]

set size ratio -1

set samples 256

set isosamples 256

set contour base

set cntrparam levels discrete 0.0

unset key

unset surface

set table 'f1.dat'

splot f1(x,y)

unset table

set table 'f2.dat'

splot f2(x,y)

unset table

set table 'f3.dat'

splot f3(x,y)

unset table

set table 'f4.dat'

splot f4(x,y)

unset table

set table 'f5.dat'

splot f5(x,y)

unset table

set table 'f6.dat'

splot f6(x,y)

unset table

plot 'f1.dat' w l lw 2 lc rgb "#0099ff", \

       'f2.dat' w l lw 2 lc rgb "#0099ff", \

       'f3.dat' w l lw 2 lc rgb "#0099ff", \

       'f4.dat' w l lw 2 lc rgb "#0099ff", \

       'f5.dat' w l lw 2 lc rgb "#0099ff", \

       'f6.dat' w l lw 2 lc rgb "#0099ff"

Example 3.

(n = 8, ratio = 1)

Download Graph files:

• folding_stars.zip

( Mathematical softwares used: Graph, gnuplot )


Related posts:

• Folding Star Polygons using Equations