Folding Star Polygons using Equations (用方程式摺星形多邊形)

Regular star polygon {8/3}

Regular star polygon {12/4}

Regular star polygon {12/5}

Regular star polygon {10/4}

Regular star polygon {15/6}

( Mathematical softwares used: Graph, gnuplot )


Related posts:

• Folding Stars using Equations
• Regular Polygons and Regular Star Polygons

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

Momotaro Peach Boy (桃太郎)

Inequalities:

(x² + 4(y+1)²) [(x² + y² - 3)sqrt(x² + y²) - a(1 - by/sqrt(x² + y²))^ln(c + abs(x)/sqrt(x² + y²))] < d

a = 5, -1.25, -2.1, -2.2, -2.36, -2.6

b = 0.84,  c = 0.5,  d = 0.1

( Softwares used: Graph, PhotoScape )

Lion Man

Equations:

(x² + y² - 3) sqrt(x² + y²)

= a(1 + sign(a)by/sqrt(x² + y²))^ln(c + abs(x)/sqrt(x² + y²))

Examples:

Download page

Weird Face - a family of implicit polar equations

( Softwares used: gnuplot, PhotoScape )

Face Slimming

The main equations are:

r³ - 3r + 1.95 = 4 abs(sinθ)^(c + sinθ)

(c = 1, 2, 3, ..., 11)

( Softwares used: gnuplot, PhotoScape )

Butt Head Girls

The main equation is a branch of the polar equation

r³ - 3r + 4 = a(1 + sinθ)^abs(cosθ), (where 4 ＜ a ＜ 4.1).

( Mathematical software used: gnuplot )