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Math Help - Finding a line/ trapezoid intersection through system of equations

  1. #1
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    Finding a line/ trapezoid intersection through system of equations

    Hello, I have a problem that I so far haven't been able to solve, I know it includes geometry but I need to find a way to solve it looking at it as functions, a system of linear equations or such that might be solved in a matrix. I want to find out if a line( finite) is either inside or intersecting a trapezoid. I know that the line can never be intersecting the trapezoid twice and the line is always parallel to the parallel lines of the trapezoid. I have tried to find an approach using a parametrized equation of the plane and line intersection but I dont think it covers all cases.
    ( something like this ( from wikipedia) : la+(lb+la)t=p0+(p1-p0)u+(p2-p0)v

    thanks
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  2. #2
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    Here's an idea.

    1. Your test line is parallel to the parallel lines of the trapezoid. Hence, there's no need to test for intersection with the parallel lines.
    2. Parametrize the non-parallel lines of the trapezoid as follows:

    y_{1}=m_{1}t+b_{1},\quad t\in[a_{1},b_{1}], and

    y_{2}=m_{2}t+b_{2},\quad t\in[a_{2},b_{2}].

    Parametrize your test line in the same way:

    z=m_{z}t+b_{z},\quad t\in[a_{z},b_{z}].

    3. Attempt to solve, one after the other, the equations z=y_{1}, and z=y_{2} for t. In both cases, there should be exactly one solution, since the lines are obviously not parallel. Test the value of t and see if it is inside both allowable ranges. For example, if you're solving z=y_{1}, then make sure both that t\in[a_{1},b_{1}] and t\in[a_{z},b_{z}]. If so, then you have an intersection. Otherwise, you don't.

    Make sense?
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  3. #3
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    Thanks! a question though, what is "a" in the interval?
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  4. #4
    A Plied Mathematician
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    a_{1} is the left-hand boundary for the interval [a_{1},b_{1}]. For example, if

    [a_{1},b_{1}]=[2,5], then a_{1}=2 and b_{1}=5.

    Does that make sense?
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  5. #5
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    Just browsing by to say a thanks a lot . The solution is great, working with wind turbine wakes here.
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  6. #6
    A Plied Mathematician
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    Great! You're very welcome for any help I could give. Have a good one!
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