Re: Insight into Solve...

• To: mathgroup at smc.vnet.net
• Subject: [mg89838] Re: [mg89815] Insight into Solve...
• From: bsyehuda at gmail.com
• Date: Sun, 22 Jun 2008 03:23:43 -0400 (EDT)
• References: <200806210931.FAA15784@smc.vnet.net>

```Hi David,
not automatic as I would like, but after the first FullSimplify you can use
a replacement rule
... /. (x1-x2)^2+(y1-y2)^2->d^2, and the Fullsimplify again with all the
previous assumptions with the addition 0<=d<=2r
you will notice that symmetry is more apparent
I hope this helps

p.s.
Appart on the last expression (after second FullSimplify) may be even
clearer

yehuda

On Sat, Jun 21, 2008 at 12:31 PM, 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?
>