## Sunday, August 21, 2016

### Tiling using Rounded Crosses

( Mathematical software used: Graph )

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## Wednesday, August 17, 2016

### Rounded Cross - cross with rounded corners (圓角十字形)

gnuplot script:

a = 2
b = 1
c = 1
w = 8
max(x,y) = x < y ? y : x
min(x,y) = x < y ? x : y
fun(x,y) = \
min(min(max(abs(x)-b,0)**2 + max(abs(y)-(a+b+2*c),0)**2 - c**2, \
max(abs(x)-(a+b+2*c),0)**2 + max(abs(y)-b,0)**2 - c**2), \
max((abs(x)-(b+c))**2 + (abs(y)-(b+c))**2 - c**2, \
c**2 - (abs(x)-(b+2*c))**2 - (abs(y)-(b+2*c))**2))
set xrange [-w:w]
set yrange [-w:w]
set size ratio -1
set samples 512
set isosamples 512
set contour base
set cntrparam levels discrete 0.0
unset key
unset surface
set table 'fun.dat'
splot fun(x,y)
unset table
plot 'fun.dat' w l lw 1.5 lc rgb "red"

(a = 2, b = 1, c = 1)

(a = 4, b = 0.5, c = 0.5)

(a = 0, b = 0, c = 2)

(a = 0, b = 2, c = 4/3)

(a = 2, b = 0, c = 4/3)

( Mathematical softwares used: Graph, gnuplot )

Related posts:

## Sunday, August 7, 2016

### Inequalities (Relations) for the Olympic Rings:

1. (Blue Ring)  min(max(R1(x,y), -y), max(R1(x,y), y, -R4(x,y))) < 0
2. (Black Ring)  min(max(R2(x,y), -y, -R4(x,y)), max(R2(x,y), y, -R5(x,y))) < 0
3. (Red Ring)  min(max(R3(x,y), -y, -R5(x,y)), max(R3(x,y), y)) < 0
4. (Yellow Ring)  min(max(R4(x,y), -y, -R1(x,y)), max(R4(x,y), y, -R2(x,y))) < 0
5. (Green Ring)  min(max(R5(x,y), -y, -R2(x,y)), max(R5(x,y), y, -R3(x,y))) < 0

Functions:
• R1(x,y) = (sqrt((x+2d)² + (y-h)²) - a)² - b²
• R2(x,y) = (sqrt(x² + (y-h)²) - a)² - b²
• R3(x,y) = (sqrt((x-2d)² + (y-h)²) - a)² - b²
• R4(x,y) = (sqrt((x+d)² + (y+h)²) - a)² - b²
• R5(x,y) = (sqrt((x-d)² + (y+h)²) - a)² - b²

Constants:
• a = 49.25
• b = 4.75
• d = 58.5
• h = 25