Re: [Algorithms] 3d Lines Intersection

[Stephen J B. <sj...@li...>]

On Sun, 30 Jul 2000, Ron Levine wrote:

> Jim Offerman  wrote:
> 
> >> Is there a good algo for finding a point of intersection between two *3D
> >lines*?
> >> The lines is specified by their point (x,y,z) and their direction
> >(dx,dy,dz)
> >
> >When I wrote my first renderer, I used a simple 3D intersection algorithm in
> >my (very alternative) projection system:
> >
> >- calculate (x, y) intersection (result <x1, y>)
> >- calculate (x, z) intersection (result <x2, z>)
> >- if (x1 == x2) the lines intersect at <x1, y, z>
> >
> >Of the top of my head I can't say whether this algorithm is 100% correct
> 
> It is my sad duty to have to inform you (and I dearly hope that the
> news causes no undue emotional stress), that this putative algorithm
> is wrong.  
> 
> Here is the clearest, simplest to understand counterexample that I
> could construct (and there are clearly a bazillion others).
> 
> Let line 1 be the x axis, so the line with parametric representation
> (s, 0, 0)   
> 
> Let line 2 be the line with parametric representation (0, t, t+1).
> Thus line 2 lies in the yz plane where it has the implicit equation 
> z = y+1.
> 
> The projection of line 1 on the xy plane is just the x axis.
> The projection of line 2 on the xy plane is the y axis,
> The intersection of these projections is (0,0,0).
> 
> The projection of line1 on the xz plane is the x axis
> The projection of line 2 on the xz plane is the z axis
> The intersection of these projections is (0,0,0)
> 
> So in your notation  x1 = 0 = x2,  and by your algorithm you claim
> that the lines intersect at (0,0,0).  In fact they do not.  Line 2
> does not even pass through the point (0,0,0),  In fact these two lines
> do not intersect at all, ever, nowhere, no way.  In fact, it is a
> problem in freshman analytic geometry to show that the distance of
> closest approach of these two lines is sqrt(1/2), but I could as
> easily have made it a billion units. 
> 
> > (I
> >am not a math guy...),
> 
> Which is no license to post wrong stuff..
> 
> > but, if properly implemented, it will certainly be
> >fast and provide at least a reasonable estimation.
> >
> 
> Bah.
> 
> I'm growing weary of playing math cop.  Soon gonna turn in my badge.
> 
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> 

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