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Re: Transformstion to canonical form
*To*: mathgroup at smc.vnet.net
*Subject*: [mg65504] Re: Transformstion to canonical form
*From*: dh <dh at metrohm.ch>
*Date*: Wed, 5 Apr 2006 06:56:06 -0400 (EDT)
*References*: <e0j48b$g7l$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Hi Wolfgang,
(u,v,w) describe an arbitrary circle, whereas (G,b) describe a circle
through the origin. Therefore, an additional equation must exist, namely:
w=u^2+v^2
To make life a bit easier (without loss of generality), I would advice
to use a symmetrical arrangement, z.B.
x1 = {-b, 0}; x2 = {b, 0}; x0 = {x, y};
This makes u=0, what leaves only v and w
With this we get (I square to get rid of square roots
):
eq1 = ((x0-x1).(x0-x2))^2==G^2 (x1-x0).(x1-x0) (x2-x0).(x2-x0);
eq2 = x^2 + (y - v)^2 == w^2;
Reduce will then give quit a lot of possible solutions:
Reduce[{eq1, eq2}, {v, w}, {x, y}]
on of which e.g.:
{w -> -(v/G), v -> (b*G)/Sqrt[1 - G^2]}
where G^2-1=!=0 and G=!= 0
Daniel
Dr. Wolfgang Hintze wrote:
> The following rather simple problem led my to some genral questions of
> how to proceed in symbolic transformations which I would do on paper
> within Mathematica. I'm sure you can give useful hints which I apreciate
> in advance.
>
> Consider a slight generalization of the Thales circle: In a triangle
> ABC, let g be the fixed angle in vertex C. (In the Thales case we have g
> = Pi/2).
>
> What is the equation for the coordinates (x,y) of the point C?
>
> The immediate statement is derived from the scalar product of the two
> vectors (C-A) and (C-B):
>
> (1)
> (C-A).(C-B)== Sqrt[(C-A).C-A)] Sqrt[(C-B).(C-A)]
>
> where
>
> (2)
> G = Cos[g]
>
> Putting
>
> A={0,0}; B={b,0};C={x,y} we obtain the equation:
>
> eq1 = x^2 - b*x + y^2 == G*Sqrt[x^2 + y^2]*Sqrt[(x - b)^2 + y^2]
>
> Now: what is a useful procedure in Mathematica to transform eq1 into an
> equation of the form
>
> eq2 = (x - u)^2 + (y - v)^2 == w
>
> and to find the parameters u, v, and w in terms of b and G?
>
> Regards,
> Wolfgang
>
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