       Help with geometry problem required.

• To: mathgroup at smc.vnet.net
• Subject: [mg20801] Help with geometry problem required.
• From: Michael Ellis <michael at digsci.demon.co.uk>
• Date: Sun, 14 Nov 1999 18:13:51 -0500 (EST)
• Organization: Digital Scientific
• Sender: owner-wri-mathgroup at wolfram.com

```I am new to this news group so please forgive me if this is an
inappropriate posting.

My Problem:
I have three points marked on a piece of rigid card at positions p1, p2
and p3. The card is moved, by translation and or rotation, but not
otherwise distorted to a new location. The three points p1, p2 and p3
are now at new positions say P1, P2 and P3. My question: Is there a 4 by
4 Transform M that uniquely describes this relocation and if so how can
I derive M given p1, p2, p3, P1, P2 and P3.

My first and naive attempt was as follows:
--------------
p1 = {x1, y1, z1, 1};
p2 = {x2, y2, z2, 1};
p3 = {x3, y3, z3, 1};

P1 = {X1, Y1, Z1, 1};
P2 = {X2, Y2, Z2, 1};
P3 = {X3, Y3, Z3, 1};

M = {{a, b, c, d}, {e, f, g, h}, {i, j, k, l}, {0, 0, 0, 1}};

DistanceSq[{x1_, y1_, z1_, 1}, {x2_, y2_, z2_, 1}] :=
(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2;

Simplify[
Solve[
{M.p1 == P1, M.p2 == P2, M.p3 == P3,
DistanceSq[p1, p2] == DistanceSq[P1, P2],
DistanceSq[p1, p3] == DistanceSq[P1, P3],
DistanceSq[p3, p2] == DistanceSq[P3, P2]},
{a, b, c, d, e, f, g, h, i, j, k, l}]
]
------------------------
This yields: "Equations may not give solutions for all \"solve\"
variables."

I have used the fact that the distance between the three points remains
unchanged during the relocation. I suspect that there are some other
constraints that I could be using given that M is restricted to rotation
and translation only.

Michael Ellis - Technical Director - Digital Scientific Ltd. UK

Please copy any replies to my email: michael at digsci.demon.co.uk

```

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