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Re: Insight into Solve...
*To*: mathgroup at smc.vnet.net
*Subject*: [mg89835] Re: [mg89815] Insight into Solve...
*From*: DrMajorBob <drmajorbob at att.net>
*Date*: Sun, 22 Jun 2008 03:23:08 -0400 (EDT)
*References*: <18655235.1214044423275.JavaMail.root@m08>
*Reply-to*: drmajorbob at longhorns.com
This helps a bit, I think:
Solve[{(x1 - x0)^2 + (y1 - y0)^2 == r^2, (x2 - x0)^2 + (y2 - y0)^2 ==
r^2}, {x0, y0}] // Simplify
or this:
Solve[{(x1 - x0)^2 + (y1 - y0)^2 == (x2 - x0)^2 + (y2 - y0)^2}, {x0}]
or:
{p0, p1, p2} = {{x0, y0}, {x1, y1}, {x2, y2}};
midPt = Mean@{p1, p2};
Solve[(p0 - midPt).(p1 - p2) == 0, x0] // ExpandAll //
Collect[#, y0] &
The last two options don't take account of the known radius, so they
return a line of possible centers.
Bobby
On Sat, 21 Jun 2008 04:31:17 -0500, David Reiss <dbreiss at gmail.com> wrote:
> Some insight requested...:
>
> When Solving for the center of a circle that goes through two
> specified points,
>
> Solve[{(x1 - x0)^2 + (y1 - y0)^2 == r^2, (x2 - x0)^2 + (y2 - y0)^2 ==
> r^2}, {x0, y0}]
>
> the result gives expressions for x0 and y0 that are structurally very
> different even though the symmetry of the problem says that they are,
> in fact expressions that are ultimately very similar.
>
> My question is what is the reason in the algorithm that Solve uses
> that causes the initial results to structurally look so different. It
> appears that Solve is not aware of the symmetry in the problem.
>
> Note that if instead of using x0 and y0 one used z0 and y0, then the
> structural forms of the expressions are reversed suggesting that Solve
> is taking variables alphabetically (no surprise here).
>
> The problem with this sort of result from Solve is that one needs to
> explicitly manipulate the resulting expressions to exploit the
> symmetries in order for the final expressions to structurally/visually
> exhibit those symmetries. FullSimplify does not, starting from the
> results of Solve, succeed in rendering the final expressions into the
> desired form.
>
> E.g, if temp is the result of the Solve command above,
>
> FullSimplify[circlePoints,
> Assumptions -> {r >= 0, x1 \[Element] Reals, x2 \[Element] Reals,
> y1 \[Element] Reals, y2 \[Element] Reals}]
>
> does not sufficiently reduce the expressions in to forms that
> explicitly exhibit the symmetry transform into one another.
>
> Is there another approach that comes to anyones' mind that will simply
> yield the anticipated results?
>
> Thanks in advance,
>
> David
>
>
--
DrMajorBob at longhorns.com
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