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Find closest points between lines

  1. May 26, 2010 #1
    1. The problem statement, all variables and given/known data
    I have two lines :
    a,u,b,v are vectors.

    [tex]A=\left\{a+s*u|s \in R \right\} B = \left\{b+t * v|t \in R \right\}[/tex]

    The two lines does not touch each other (does not meet)
    I need to find the closest point between the lines.


    2. Relevant equations



    3. The attempt at a solution

    I know several ways, But all of them are giving me unbelivable long functions..
    There must be a short way.
    One options it to build a vector between two random points in the lines and then the scalar multipltion of them need to give me 0 .
    a,u,b,v are vectors.
    [tex](b+t*v-a-s) \bullet v = 0[/tex]
    [tex](b+t*v-a-s) \bullet u = 0[/tex]
    But as I said I tried to solved it and it got to be very very very long and I always made errors...

    Second way it to find [tex]u \times v [/tex] this is a vector that is vertical to both lines so if I need to fins the solution of :
    [tex]b+t*v+q(u \times v) = a+s*u [/tex]
     
  2. jcsd
  3. May 26, 2010 #2

    vela

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    Let me rewrite your last equation a bit.

    [tex]q(\textbf{u} \times \textbf{v}) = (\textbf{a}+s\textbf{u})-(\textbf{b}+t\textbf{v})[/tex]

    The RHS corresponds to the vector beginning on a point on B and ending on a point on A. Now try taking the dot product of both sides with [itex]\textbf{u} \times \textbf{v}[/itex]. What geometrically does that correspond to?
     
  4. May 27, 2010 #3
    Thank you for your respond.
    English is not my first language and
    Sadly I am not sure I understand what you mean in "try taking the dot product of both sides with [itex]\textbf{u} \times \textbf{v}[/itex]"
    Do you mean that I need to build three equation :
    First we know that [itex]\textbf{u} \times \textbf{v} = (u_{2}v_{3}-v_{2}u_{3} ,-u_{1}v_{3}+v_{1}u_{3} , u_{2}v_{1}-v_{2}u_{1} )[/itex]
    After we found the vector we can build 3 equations .. Is this is what you meant I need to do>?, [itex] q(u_{2}v_{3}-v_{2}u_{3}) = a_{1}+s*u_{1}-b_{1}-t*v_{1}[/itex]<-- Something like this? this is the first equation
    Because I did it and it didnt really gave me anything
     
  5. May 27, 2010 #4

    vela

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    No, that's not what I meant. "Dot product" is another way of saying "scalar product," so I was saying you should do this:

    [tex]q(\textbf{u} \times \textbf{v})\cdot(\textbf{u} \times \textbf{v}) = [(\textbf{a}+s\textbf{u})-(\textbf{b}+t\textbf{v})]\cdot(\textbf{u} \times \textbf{v})[/tex]
     
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