Wednesday, June 22, 2016

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:

Friday, June 17, 2016

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)

Graph files:



( Mathematical softwares used: Graph, gnuplot )


Related posts:

Sunday, June 12, 2016

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 )

Friday, June 10, 2016

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


Sunday, June 5, 2016

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 )

Thursday, June 2, 2016

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 )