Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!tut.cis.ohio-state.edu!ucbvax!agate!eos!jbm
From: jbm@eos.UUCP (Jeffrey Mulligan)
Newsgroups: comp.graphics
Subject: Re: Tangents to three circles (SPOILER)
Message-ID: <4826@eos.UUCP>
Date: 23 Aug 89 19:33:42 GMT
References: <11391@polya.Stanford.EDU> <3828@internal.Apple.COM>
Organization: NASA Ames Research Center, California
Lines: 58

OK, how about this:

First consider the case of just two tangent circles:
circle X with radius r, and circle A with radius a.
Let d be the distance between the centers.
There are 3 cases:

1.  A is inside of X:			r = d + a
2.  X is inside of A:			r = a - d
3.  A and X are externally tangent:	r = d - a

Now we impose the condition that circle X must be tangent to
a second circle B with radius b.  We will rename the distance
between the centers of X and A (previously "d") "da", and
call the distance between the centers of X and B "db".
Now we can write down the same 3 cases for B and X,
and equate r's for A and B, for a total of nine cases:

1 (1,1)	da + a = db + b			da - db = b - a
2 (1,2)	da + a = b - db			da + db = b - a
3 (1,3)	da + a = db - b			da - db = -(a + b)
4 (2,1)	a - da = db + b			da + db = a - b
5 (2,2)	a - da = b - db			da - db = a - b
6 (2,3)	a - da = db - b			da + db = a + b
7 (3,1)	da - a = db + b			da - db = a + b
8 (3,2)	da - a = b - db			da + db = a + b
9 (3,3)	da - a = db - b			da - db = a - b

We can group these cases:

I	6,8	da + db = a + b		ellipse
II	2,4	da + db = | a - b |	ellipse
III	1,5,9	da - db = +- ( a - b )	hyperbola
IV	3,7	da - db = +- ( a + b )	hyperbola

These are all conics having foci at the centers of A and B.
For any point on one of these curves, there is a corresponding
r for which the circle centered at the chosen point with radius r
will be tangent to A and B.

Note that for the case of A and B concentric that da=db regardless
of r, so there are no solutions to cases III and IV (well, unless
a==b).

Now if we add a third circle C, we get four more conic sections
(defined bewteen A and C or B and C).  Any intersections between
the new four and the original four should correspond to valid
solutions.

It is left as an exercise to the reader to show that the max. number
of solutions is 9 or whatever it is.


-- 

	Jeff Mulligan (jbm@aurora.arc.nasa.gov)
	NASA/Ames Research Ctr., Mail Stop 239-3, Moffet Field CA, 94035
	(415) 694-6290