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)

Infinite Families of Concentric Circles (1) (無限同心圓系 1)

Type 1.

Annuluses formed by an infinite family of concentric circles

in geometric progression with common ratio r (0 < r < 1).

Equations of the circles:

x² + y² = (1 + r + r² + r³ + ... + r^k)², k=0,1,2,3,....

Regions determined by inequalities formed by five infinite families

of concentric annuluses.

r = 1 / golden ratio = 2 / (1+√5) ~ 0.618

h = 2  (h is the displacement parameter.)

r = 0.75, h = 2.5

r = 0.8, h = -3

r = 1 / golden ratio, h = 1.5

r = 0.75, h = 2.5

Type 2.

Annuluses formed by an infinite family of concentric circles

in geometric progression with common ratio r (0 < r < 1).

Equations of the circles:

x² + y² = (r^k)², k=0,1,2,3,....

r = 0.5, h = 0.25

r = 0.8, h = 0.5

r = 0.999, h = 1

r = 0.8, h = 0.5

Type 3.

Annuluses formed by two infinite families of concentric circles.

r1 = the common ratio of the outer family  (0 < r1 < 1)

r2 = the common ratio of the inner family  (0 < r2 < 1)

r1 = r2 = 0.5, h = 1.5

r1 = 0.65, r2 = 0.8, h = 2.5

r1 = 1 / golden ratio, r2 = 0.75, h = 2

r1 = r2 = 0.5, h = 1.5

( Mathematical software used: Graph )

Related posts:

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