Friday, December 6, 2013

Infinite Families of Tangent Circles (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 )

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Wednesday, November 27, 2013

Infinite Families of Tangent Circles (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 )

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Friday, November 1, 2013

Infinite Families of Concentric Circles (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 )

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