Five Interlaced Moons

Five Interlaced Crescents

Generalization:

Let A and B be two disks with radii a and b.

The set difference A\B is used to plot the moons 

in the following animations.

Animation parameter: the position of B

Animation parameter: b (the radius of B)

Animation parameter: the position of A

Animation parameter: a (the radius of A)

( Mathematical softwares used: gnuplot, GeoGebra )

Four Interlaced Crescents

A cross pattée!

( Mathematical softwares used: Graph, gnuplot )


Related posts:

• Cross Patty (cross pattée)

Triple Crescent Moon

Triple Crescent Moon Symbol

(Boundary curves: circles)

Boundary curves: superellipses with exponent 2.5

Boundary curves: superellipses with exponent 4

A monster face!

Hi!

Exponents: 1.5, 2, 2.5, 3, 3.5, 4

( Mathematical software used: gnuplot )

How to Draw the Olympic Rings Mathematically?

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

Download Examples:

• olympic_rings.zip

( Mathematical softwares used: Graph, gnuplot )

Sinusoidal Strings - made from a ring of circles

Sinusoidal Strings of Pebbles (r=0.1, n=32)

(r=0.15, n=42)

(r=0.4, n=60)

A ring of circles can be expressed as follows:
x = cos(t-mod(t,s)) + r*cos(n*mod(t,s))
y = sin(t-mod(t,s)) + r*sin(n*mod(t,s))
r=0.1, n=32, s=2*pi/n, t=0...2*pi.

Note that mod(t,s) = t - s*floor(t/s).

gnuplot Examples:

Method 1:
r = 0.1
n = 32
s = 2*pi/n
f(t) = cos(s*floor(t/s)) + r*cos(n*(t-s*floor(t/s)))
g(t) = sin(s*floor(t/s)) + r*sin(n*(t-s*floor(t/s)))
theta(t) = atan2(g(t), f(t))
set terminal wxt enhanced font "Arial, 10"
set xtics ('-2π' -2*pi, '-π' -pi, '0' 0, 'π' pi, '2π' 2*pi)
set xrange [-2*pi-0.4:2*pi+0.4]
set yrange [-pi-0.2:pi+0.2]
set size ratio -1
set samples 10000
set multiplot
set parametric
unset key
do for [k=-2:2:2]{
  plot [-pi:pi] theta(t) + k*pi, f(t) w p pt 0.5 lc rgb "#0099cc"
}
do for [k=-2:2:2]{
  plot [-pi:pi] theta(t) + k*pi, g(t) w p pt 0.5 lc rgb "#ff00ff"
}
unset multiplot


Method 2:
r = 0.1
n = 32
s = 2*pi/n
f(t) = cos(s*floor(t/s)) + r*cos(n*(t-s*floor(t/s)))
g(t) = sin(s*floor(t/s)) + r*sin(n*(t-s*floor(t/s)))
angle(t) = atan2(g(t), f(t))
theta(t) = angle(t - 2*pi*floor((t+pi)/(2*pi))) + 2*pi*floor((t+pi)/(2*pi))
set terminal wxt enhanced font "Arial, 10"
set xtics ('-2π' -2*pi, '-π' -pi, '0' 0, 'π' pi, '2π' 2*pi)
set xrange [-2*pi-0.4:2*pi+0.4]
set yrange [-pi-0.2:pi+0.2]
set size ratio -1
set samples 10000
set parametric
unset key
plot [-4*pi:4*pi] theta(t), f(t) w p pt 0.5 lc rgb "#0099cc", \
                      theta(t), g(t) w p pt 0.5 lc rgb "#ff00ff"

The result plot:

Graph Examples:

• Method 1 ( sinusoidal_strings_01.grf )
• Method 2 ( sinusoidal_strings_02.grf )

( Mathematical softwares used: Graph, gnuplot )

Infinite Families of Tangent Circles (3) (無限相切圓系 3)

Type 1.

Equations of the Circles:

(x + a(1 - p^ceil(k/2)))² + (y + b(1 - p^floor(k/2)))² = d^k

a = (1 - r)/(1 + r²)

b = r(1 - r)/(1 + r²)

d = r²

p = -r²

0 < r < 1

k = 0, 1, 2, 3, ....

Type 2.

Equations of the Circles:

(x + a(1 - p^floor(k/2)))² + (y + b(1 - p^ceil(k/2)))² = d(1 - r^(k+1))²

a = r²/(1 + r²)

b = r/(1 + r²)

d = (1/(1-r))²

p = -r²

0 < r < 1

k = 0, 1, 2, 3, ....

Type 3.

Equations of the Circles:

(See the equations for Type 1 and Type 2.)

( Mathematical software used: Graph )

Related posts:

• Infinite Families of Tangent Circles (1)
• Infinite Families of Tangent Circles (2)
• Infinite Families of Concentric Circles (1)
• Infinite Families of Concentric Circles (2)

Infinite Families of Tangent Circles (2) (無限相切圓系 2)

Type 1.

r = common ratio = 3/4

h = displacement = 0

r = 3/4, h = 1

r = 3/4, h = 0.5

Type 2.

r = 3/4, h = 0

r = 3/4, h = 4

r = 4/5, h = 0.5

Type 3.

r1 = 4/7, r2 = 3/4, h = 0

r1 = 4/7, r2 = 3/4, h = 4

r1 = 4/7, r2 = 3/4, h = 3

r1 = 4/7, r2 = 3/4, h = 3.2

r1 = 5/9, r2 = 4/5, h = 0.5

r1 = 4/7, r2 = 3/4, h = 0.8 ~ 3.2

( Mathematical software used: gnuplot )

Related posts:

• Infinite Families of Tangent Circles (1)
• Infinite Families of Tangent Circles (3)
• Infinite Families of Concentric Circles (1)
• Infinite Families of Concentric Circles (2)

Infinite Families of Concentric Circles (2) (無限同心圓系 2)

Type 1.

Type 2.

Type 3.

( Mathematical software used: Graph )

Related posts:

• Infinite Families of Concentric Circles (1)
• Infinite Families of Tangent Circles (1)
• Infinite Families of Tangent Circles (2)
• Infinite Families of Tangent Circles (3)