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Topic: quickest way of finding intersection of 2 planes
Replies: 56   Last Post: Jun 19, 2000 7:04 PM

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 Steve Lord Posts: 611 Registered: 12/6/04
Re: quickest way of finding intersection of 2 planes
Posted: Jun 9, 2000 10:55 PM

On Fri, 9 Jun 2000 necm500@yahoo.com wrote:

> In article <3940C872.5C6D@pointecom.net>,
> killbeck@pointecom.net wrote:

> > necm500@yahoo.com wrote:
> > >
> > > Suppose I have 2 non-parrellel planes represented by a_1*x + b_1*y +
> > > c_1*z + d_1 = 0 and a_2*x + b_2*y + c_2*z + d_2 = 0, respectively.
> > > Since they're not parrellel, their intersection would be a line.

> What's
> > > the quickest way to find that intersection line as represented in
> > > parametric form, or implicit form?
> > >
> > > Thanks a lot for your help.
> > >
> > > Sent via Deja.com http://www.deja.com/
> > > Before you buy.

> >
> > Cross-product of their normals comes to mind.

>
> 2 questions:
>
> 1. why is the cross product parallel to the intersection line? Is there
> a quick, short and intuitive reason or does one have to go through some
> kind of formal proof?

Sketch out two planes. Draw normals to each of them, and have them
intersect at a point. You can do this (why?) Now, consider what the
cross product of those two vectors will give you (especially what the
angles between the normals and the cross product are) and see if you
believe that the cross product is parallel to the intersection line.

You can formally prove it, but see if you can intuit it first.

> 2. Given the vector of cross-product, I still need to find a point on
> the intersection line in order to have a line equation. How should I go

Pick an x. Solve for y in terms of z in one of the equations. Plug and
chug in the other.

Steve L