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the Problem of Apollonius:Calculating (x,y,r) of eight circles given 3 circles
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BookWight
May 12, 2013, 3:37:25 PM5/12/13
to
Hi -
Is it possible to calculate the (x,y) co-ordinates and the radius of each
of the eight circles, given three circles of known position and radius?
original 3 circles:
(ax1,ay1,ar1)
(ax2,ay2,ar2)
(ax3,ay3,ar3)
solutions:
(sx1,sy1,sr1)
(sx2,sy2,sr2)
etc
does it help to assume all 3 circles are in the 1st quadrant?
Is it possible to calculate the (x,y) co-ordinates and the radius of each
of the eight circles, given three circles of known position and radius?
original 3 circles:
(ax1,ay1,ar1)
(ax2,ay2,ar2)
(ax3,ay3,ar3)
solutions:
(sx1,sy1,sr1)
(sx2,sy2,sr2)
etc
does it help to assume all 3 circles are in the 1st quadrant?
William Elliot
May 12, 2013, 9:54:30 PM5/12/13
to
On Sun, 12 May 2013, BookWight wrote:
> Is it possible to calculate the (x,y) co-ordinates and the radius of
> each of the eight circles, given three circles of known position and
> radius?
>
No, it's not possible to know anything about the other five circles.
> Is it possible to calculate the (x,y) co-ordinates and the radius of
> each of the eight circles, given three circles of known position and
> radius?
>
> original 3 circles:
>
> (ax1,ay1,ar1)
> (ax2,ay2,ar2)
> (ax3,ay3,ar3)
>
> solutions:
>
> (sx1,sy1,sr1)
> (sx2,sy2,sr2)
> etc
>
> does it help to assume all 3 circles are in the 1st quadrant?
>
a capital letter.
Christopher J. Henrich
May 13, 2013, 2:53:23 PM5/13/13
to
In article <XnsA1BE806A3...@216.151.153.167>, BookWight
books on "advanced geometry" or "college geometry" discuss this. Also,
see /What/ /Is/ /Msthematics/ by Courant and Robbins, recently reissued
with added comments by Ian Stewart.
--
Chris Henrich
http://www.mathinteract.com
God just doesn't fit inside a single religion.
<Book...@publicspam.com> wrote:
> Hi -
>
> Is it possible to calculate the (x,y) co-ordinates and the radius of each
> of the eight circles, given three circles of known position and radius?
Yes, because the problem is solvable with ruler & compass. Various
> Hi -
>
> Is it possible to calculate the (x,y) co-ordinates and the radius of each
> of the eight circles, given three circles of known position and radius?
books on "advanced geometry" or "college geometry" discuss this. Also,
see /What/ /Is/ /Msthematics/ by Courant and Robbins, recently reissued
with added comments by Ian Stewart.
>
> original 3 circles:
>
> (ax1,ay1,ar1)
> (ax2,ay2,ar2)
> (ax3,ay3,ar3)
>
> solutions:
>
> (sx1,sy1,sr1)
> (sx2,sy2,sr2)
> etc
>
> does it help to assume all 3 circles are in the 1st quadrant?
I don't think so.
> original 3 circles:
>
> (ax1,ay1,ar1)
> (ax2,ay2,ar2)
> (ax3,ay3,ar3)
>
> solutions:
>
> (sx1,sy1,sr1)
> (sx2,sy2,sr2)
> etc
>
> does it help to assume all 3 circles are in the 1st quadrant?
--
Chris Henrich
http://www.mathinteract.com
God just doesn't fit inside a single religion.
Ken Pledger
May 13, 2013, 5:40:30 PM5/13/13
to
In article <130520131453237591%chen...@monmouth.com>,
The problem goes back to Apollonius. Courant & Robbins give an
elegant treatment by inversion, but their earlier discussion using
coordinates just explains why it can be done in principle, without
actually doing it.
There's an interestng little passage about this in E.T. Bell, "Men of
Mathematics", vol.1, p.52, about Descartes and his pupil Princess
Elisabeth. On the Apollonius circle problem, Bell says "This problem is
a fine specimen of the sort that are _not_ adapted to the crude brute
force of elementary Cartesian geometry. _Elisabeth solved it by
Descartes' methods._ It was rather cruel of him to let her do it...."
Ken Pledger.
"Christopher J. Henrich" <chen...@monmouth.com> wrote:
> In article <XnsA1BE806A3...@216.151.153.167>, BookWight
> <Book...@publicspam.com> wrote:
> ....
> In article <XnsA1BE806A3...@216.151.153.167>, BookWight
> <Book...@publicspam.com> wrote:
> > Is it possible to calculate the (x,y) co-ordinates and the radius of each
> > of the eight circles, given three circles of known position and radius?
> Yes, because the problem is solvable with ruler & compass. Various
> books on "advanced geometry" or "college geometry" discuss this. Also,
> see /What/ /Is/ /Msthematics/ by Courant and Robbins, recently reissued
> with added comments by Ian Stewart....
> > of the eight circles, given three circles of known position and radius?
> Yes, because the problem is solvable with ruler & compass. Various
> books on "advanced geometry" or "college geometry" discuss this. Also,
> see /What/ /Is/ /Msthematics/ by Courant and Robbins, recently reissued
The problem goes back to Apollonius. Courant & Robbins give an
elegant treatment by inversion, but their earlier discussion using
coordinates just explains why it can be done in principle, without
actually doing it.
There's an interestng little passage about this in E.T. Bell, "Men of
Mathematics", vol.1, p.52, about Descartes and his pupil Princess
Elisabeth. On the Apollonius circle problem, Bell says "This problem is
a fine specimen of the sort that are _not_ adapted to the crude brute
force of elementary Cartesian geometry. _Elisabeth solved it by
Descartes' methods._ It was rather cruel of him to let her do it...."
Ken Pledger.
Christopher J. Henrich
May 13, 2013, 7:20:31 PM5/13/13
to
In article
<ken.pledger-D344...@news.eternal-september.org>, Ken
Hmm, yes. But with the machinery of "Lie sphere geometry" (Wikipedia),
it becomes simple.
<ken.pledger-D344...@news.eternal-september.org>, Ken
it becomes simple.
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