Re:Help with geometry problem required.

• To: mathgroup at smc.vnet.net
• Subject: [mg20851] [mg20851] Re:[mg20801] Help with geometry problem required.
• From: Arnold Seiken <seikena at union.edu>
• Date: Wed, 17 Nov 1999 03:41:14 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Michael,

Here is an example which illustrates how to find the
matrix of an isometry in R3 given 3 non-collinear points
and their images:

M={{a11,a12,a13,b1},{a21,a22,a23,b2},{a31,a32,a33,b3},{0,0,0,1}};
Simplify[
Solve[
{M.{1,0,0,1} =={4,4,2,1}, M.{0,1,0,1} == { 3,3,2,1},
M.{0,0,1,1}=={4,3,3,1},

{a11,a12,a13 }.{a21,a22,a23 }==0,
{a11,a12,a13 }.{a31,a32,a33 }==0,
{a31,a32,a33 }.{a21,a22,a23 }==0,
(a11)^2 +(a12)^2 + (a13)^2 == 1,
(a21)^2 +(a22)^2 + (a23)^2 == 1,
(a31)^2 +(a32)^2 + (a33)^2 == 1,
Det[{{a11,a12,a13 },{a21,a22,a23 },{a31,a32,a33 }}]==1},

{a11,a12,a13,a21,a22,a23,a31,a32,a33,b1,b2,b3}]
]

You can then take the output of the above and input
(M/.%[[1]])//MatrixForm
to get the solution in a nice looking form.

Arnold Seiken

```

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