xyz coordinates from ortho projections

• To: mathgroup at smc.vnet.net
• Subject: [mg5147] xyz coordinates from ortho projections
• From: TTCJ34A at prodigy.com (DR JOHN C ERB)
• Date: Wed, 6 Nov 1996 01:33:17 -0500
• Sender: owner-wri-mathgroup at wolfram.com

```How can I use Mma to solve the following problem?

I have N points in 3D space, the x,y,z locations of which I do
not know, but want to find.

What I do have are the orthogonal projections of the points onto
an (x,y,z=constant) plane from a projection point (0,0,z=Zpoint),
and onto an (x=constant,y,z) plane from a projection point (x=Xpoint,
0,0).

I propose to compute a N-squared matrix of distances of closest
approach
of lines drawn from each projection point to the points on the
respective
projection planes.  I would then like to compute the sum of the
distances
of closest approach for all possible sets of lines, and use the set
with
the smallest sum as the set which represents the N points in 3D space.

Example:  For N=2  {{{P1,Q1,0.5},{P2,Q2,0.8}},
{{P1,Q2,0.8},{P2,Q1,0.9}}}
where P1 & P2 are the projection points on the first plane,
Q1 & Q2 are projection points on the second plane, and the number
represents the distance of closest approach of the lines drawn from
the respective projection points to the points on the corresponding
projection plane.

The first set adds to 0.5 + 0.8 = 1.3, while the second set
adds to 0.8 + 0.9 = 1.7; therefore, since the sum of the first set is

smaller, point P1 on the first plane
would correspond to Q1 on the second plane, and P2 to Q2.

My question is how do I determine all possible sets of points so I
can then compute the sums?

Or perhaps there is a better way of finding (x,y,z) 3D space
coordinates
from two orthogonal projections than by finding a least sum of
distances of closest approach of projection lines ?!?!?

Thanks for any help.

John C. Erb

```

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