[Haskell-cafe] Coplanarity or Colinearity [Was: low-cost matrix rank?]

[Jerzy Karczmarczuk]

Many people help Mike Meyer:

> My real problem is that I've got a list of points in R3  and want to 
> decide if they determine a plane, meaning they are coplanar but not 
> colinear. Similarly, given a list of points in R2, I want to verify 
> that they aren't colinear. Both of these can be done by converting the 
> list of points to a matrix and finding the rank of the matrix, but I 
> only use the rank function in the definitions of colinear and coplanar.
>
> Maybe there's an easier way to tackle the underlying problems. Anyone 
> got a suggestion for such?

I didn't follow this discussion, so I might have missed some essential 
issues, I apologize then. But if THIS is the problem...

All these powerful universal libraries, with several hundreds of lines 
of code are important and useful, but if the problem is to find whether 
a list of pairs (x,y) is collinear or not, I presume that such program 
as below could do. I am ashamed showing something like that...

*colin ((x,y):l) = all (\(c,d)->abs(px*d-py*c)<eps) q where
  ((px,py):q) = [(ax-x,ay-y) | (ax,ay) <- l]
*
[The iterated subtraction puts the first vector at the origin. eps is 
the precision; better avoid ==0]

For three dimensions and coplanarity, instead of separating one vector, 
we need two, in order to compute their skew product. Then, it suffices 
to verify that all its scalar products with the remaining, vanish; if 
not, they are not coplanar.

Jerzy Karczmarczuk



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