I like the idea of a non-algebraic solution too.
Here are some suggested simplifications:
load 'math/misc/amoeba'
NB. y is Cx0 Cy0 R0 Cx1 Cy1 R1 Cx2 Cy2 R2
NB. x are radius scale factors to control which circles are included
NB. in the common tangent circle. 1 to surround, _1 to exclude.
NB. returns (center, radius); final simplex volume
apollonius =: verb define"1 _
1 apollonius y
:
c=. 2{."1 y NB. centers
r=. x * {:"1 y NB. radii
v=. 1e_20 NB. goal simplex volume
dist=: r 2 : 'm + [: +/"1&.:*: n -"1 ]' c NB. distances to tangents
'soln err'=. ([: +/@:*:@, -/~...@dist) amoeba v c
if. err > 10 * v do. '' return. end. NB. no solution found
avg =. +/ % #
(, a...@dist) soln
)
Note 'Example Use'
rctst=: 0 0 1,4 0 1,:2 4 2 NB. Rosetta Code test data
(_1 _1 _1 ,: 1 1 1) apollonius rctst NB. both solutions
)
NB. Display Solutions
NB. =================
require 'graph'
NB. convert between circle definition formats
NB. (centerX,centerY,radius) to/from
NB. (topleftX,topleftY,width,height)
cr2xywh=: 2&{."1 (- ,. 2&#@:,:@+:@]) {:"1
xywh2cr=: 2&{."1 (+ ,. ]) -:@:{:"1 NB. not strictly required
draw=: [: gdellipse gddraw 0.1 * cr2xywh@,
Note 'display solutions'
draw rctst NB. initial circles
(draw _1 _1 _1&apollonius) rctst NB. internally tangent solution
(draw 1 1 1&apollonius) rctst NB. externally tangent solution
(draw ((,:-)1 1 1)&apollonius) rctst NB. both required solutions
(draw (<:+:#:i.8)&apollonius) rctst NB. all (5) solutions
)
> From: David Ward Lambert
> Sent: Sunday, 19 September 2010 02:18
>
> NB.I propose this solution. The posted solutions are algebraic.
>
> NB. amoeba is the simplex optimization method found at
> www.jsoftware.com
>
> require'amoeba'['/usr/local/j64-602/addons/math/misc/amoeba.ijs'
>
> NB. y is Cx0 Cy0 R0 Cx1 Cy1 R1 Cx2 Cy2 R2
> NB. x are radius scale factors to control which circles are included
> NB. in the common tangent circle. 1 to surround, _1 to exclude.
> NB. returns (center, radius); final simplex volume
> apollonius =: 3 :0
> 1 apollonius y
> :
> a=.9$0 0 1 NB. boolean selection vector
> c=.3 2$(-.a)#,y NB. centers
> r=.x*a#,y NB. radii
> v=.1e_20 NB. goal simplex volume
> NB. scale the problem rather than the volume?
> d=:r 2 : 'm + [: +/"1&.:*: n -"1 ]' c NB. distances to tangents
> s=.([: +/ [: *:@, -/~@:d) amoeba v c NB. solution
> avg =. +/ % #
> ((, avg@:d)&.:>@:{. , {:)s
> )
>
> NB. all 8 solutions
> (<:+:#:i.8)apollonius"1 _ ] _1 1 1,4 2 2,:1 _3 3
>
>
> NB........graphical demonstration...........
>
> load'graph'
>
> p0 =: 1,.~|:2 1 o./ 2p1 * (i. % <:) 80
>
> draw =: 4 : 0
> gdclose name=:'apollo'
> gdopen name
> gdcolor GREEN
> gdpolygon ,/"2 p0(+/ .*)("_ _1)0.1*((0 1,3 4,:6 7){,y),"2 1~(2 5
> 8{,y)*/=i.2
> gdcolor RED
> gdlines ,p0+/ .*0.1*(2{.,x),~({:,x)*=i.2
> gdshow name
> )
>
> (draw~ >@{.@:((-_1 1 1)&apollonius))_1 1 1,4 2 2,:1 _3 3
>
>
> ----------------------------------------------------------------------
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