## 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 )

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## Type 1.

r = common ratio = 3/4
h = displacement = 0

r = 3/4, h = 1

r = 3/4, h = 0.5

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 )

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## Type 3.

( Mathematical software used: Graph )

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## 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 )

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## Thursday, October 31, 2013

### Infinite Families of Tangent Circles (1) (無限相切圓系 1)

( Mathematical software used: gnuplot )

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