       Re: best line through a set of 3D points

• To: mathgroup at smc.vnet.net
• Subject: [mg18733] Re: [mg18386] best line through a set of 3D points
• From: Maarten.vanderBurgt at icos.be
• Date: Sat, 17 Jul 1999 02:36:48 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```David, Richard, Jens, Tom, Hartmut,

Thank you all very much for your help.

A solutions which was very elegant and very usefull for me came from Jens.
He suggested to fit a line through the 3 projections  on the x-y, y-x and
x-z planes. This imediately gives you the direction vector of a best fit in
3D.

David sent me a nice notebook: the essential part in there is: do not try
to find a direction vector of the type (a, b, c)  but try to find e.g. (1,
v, t) = (1, b/a, c/a), and normalize afterwards. This way  the least
squares approach by minimizing the sum of the squares of the distances from
the points to the line, also worked: you do get a set of two equations from
which you can get non-trivial solutions for v and t.

regards

Maarten

_______________________________________________________________

Maarten van der Burgt

ICOS Vision Systems
Esperantolaan 9
B-3001 Leuven, Belgium
tel. + 32 16 398220; direct + 32 16 398316; fax. + 32 16 400067
e-mail: maarten.vanderburgt at icos.be
_______________________________________________________________

Maarten.vanderBurgt at icos.be on 30-06-99 08:13:40 PM

Subject: [mg18733]  [mg18386] best line through a set of 3D points

Dear all,

This is not strictly a mathematica question but someone might have a
solution in the form of a mathematica function or so.

I have a set of 3d points, DATA = {{x1,y1,z1},{x2,y2,z3},...}, roughly
occupying a sigar shaped volume in the 3D space.
I want to find the line that best fits these points.

I tried a least squares approach: I assumed the line was going through the
average off all the points in DATA and then I tried to find a vector in the
direction of the line by minimizing the sum of the squares of the distances
from the points to the line.
For some reason I end up with a set of 3 equations which only solution is
(0,0,0).  There is probably some sensible reason for this but I did not
manage to figure out why. Maybe someone else knows?

Another approach I tried is averaging the vectors connecting each point in
DATA with the average point of DATA. This average vector could be a good
direction for the best line. But comparing this method in two dimensions
with a least squares fit, I noticed the agreement is not always good. So I
am not sure whether this is the best method.

Does anyone have a solution to this problem or can anyone point me to some
resources (book(s), web page(s)) where I could find a solution.

thanks a lot

Maarten

_______________________________________________________________

Maarten van der Burgt

ICOS Vision Systems
Esperantolaan 9
B-3001 Leuven, Belgium
tel. + 32 16 398220; direct + 32 16 398316; fax. + 32 16 400067
e-mail: maarten.vanderburgt at icos.be
_______________________________________________________________

```

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