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MathGroup Archive 2006

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Re: Finding a shorter form for coplanar point in 3D space

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65159] Re: Finding a shorter form for coplanar point in 3D space
  • From: "Valeri Astanoff" <astanoff at yahoo.fr>
  • Date: Wed, 15 Mar 2006 23:59:33 -0500 (EST)
  • References: <dv9021$nil$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

This is the way I would do it :

In[1]:=
OA={xa,ya,0};
OB={xb,yb,zb};
OC={xc,yc,0};
AB=OB-OA;
CA=OA-OC;
AD=k AB;
CD=CA+AD;
OD=OA+AD;

In[9]:=sol=First at Solve[CD.CD == l^2,k];

In[10]:=OD/.sol//Simplify

Out[10]=
{xa - ((xa - xb)*(xa^2 - xa*xb - xa*xc + xb*xc + ya^2 - ya*yb - ya*yc +
yb*yc -
      (1/2)*Sqrt[4*(xa^2 + xb*xc - xa*(xb + xc) + (ya - yb)*(ya -
yc))^2 -
         4*(-l^2 + xa^2 - 2*xa*xc + xc^2 + ya^2 - 2*ya*yc + yc^2)*(xa^2
- 2*xa*xb + xb^2 + ya^2 - 2*ya*yb + yb^2 +
           zb^2)]))/(xa^2 - 2*xa*xb + xb^2 + ya^2 - 2*ya*yb + yb^2 +
zb^2),
  ya - ((ya - yb)*(xa^2 - xa*xb - xa*xc + xb*xc + ya^2 - ya*yb - ya*yc
+ yb*yc -
      (1/2)*Sqrt[4*(xa^2 + xb*xc - xa*(xb + xc) + (ya - yb)*(ya -
yc))^2 -
         4*(-l^2 + xa^2 - 2*xa*xc + xc^2 + ya^2 - 2*ya*yc + yc^2)*(xa^2
- 2*xa*xb + xb^2 + ya^2 - 2*ya*yb + yb^2 +
           zb^2)]))/(xa^2 - 2*xa*xb + xb^2 + ya^2 - 2*ya*yb + yb^2 +
zb^2),
  (zb*(xa^2 - xa*xb - xa*xc + xb*xc + ya^2 - ya*yb - ya*yc + yb*yc -
     (1/2)*Sqrt[4*(xa^2 + xb*xc - xa*(xb + xc) + (ya - yb)*(ya - yc))^2
-
        4*(-l^2 + xa^2 - 2*xa*xc + xc^2 + ya^2 - 2*ya*yc + yc^2)*(xa^2
- 2*xa*xb + xb^2 + ya^2 - 2*ya*yb + yb^2 +
          zb^2)]))/(xa^2 - 2*xa*xb + xb^2 + ya^2 - 2*ya*yb + yb^2 +
zb^2)}


hth

v.a.


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