More on the Intersection of Two Lines in Three Dimensions
Date: 02/21/2009 at 09:00:33 From: Josemi Subject: Intersection point of Two Lines Hi, Another Dr. Math conversation shows a very interesting formula for calculating the intersection point of two lines: a(V1 x V2) = (P2 - P1) x V2 (From http://mathforum.org/library/drmath/view/63719.html ) Would you please send me the steps to obtain "a"? Your sincerely, Josemi
Date: 02/23/2009 at 08:15:00 From: Doctor George Subject: Re: Intersection point of Two Lines Hi Josemi, Look carefully at a(V1 x V2) = (P2 - P1) x V2 This is a vector equation. For there to be a solution, the i,j,k components of each side must be equal. So you could compare the i components to find the value of 'a' and then make sure this value also works for the j and k components. If you are writing computer code, I would suggest this. Take the dot product of both sides with V1 X V2 to obtain a (V1 x V2).(V1 x V2) = [(P2 - P1) x V2].(V1 x V2) or a |V1 x V2)|^2 = [(P2 - P1) x V2].(V1 x V2) This is now a scalar equation. Just divide to solve for "a" on the left hand side. Now, there is one last issue. The whole procedure assumes that the two lines intersect. The value we have for 'a' specifies a point on L1. For two lines in 3D, we must make sure that this point is also a point on L2. One way to do that is to check that the distance from this point to L2 is zero. Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/
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