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Solving trigonometric system of equations

  1. Sep 8, 2010 #1
    Hi. I'd like to know whether is it possible to do the following, and if so, how...(and also, whether is it possible to solve similar problems)

    I have parametric equation of a curve and I need to find its intersection with a ray that starts at the origin of the coordinate system and makes known angle with the positive x-axis.

    Curve:
    x = ( R-r )( cos([tex]\varphi[/tex]) - cos([tex]\theta[/tex]) ) + D cos([tex]\varphi[/tex] (R-r)/r - [tex]\theta[/tex] (R-r)/r )
    y = ( R-r )( sin([tex]\varphi[/tex]) - sin([tex]\theta[/tex]) ) + D sin([tex]\varphi[/tex] (R-r)/r - [tex]\theta[/tex] (R-r)/r )

    R, r and D are known constants.

    Ray:
    y = tan([tex]\beta[/tex]) x

    [tex]\beta[/tex], [tex]\theta[/tex] and [tex]\varphi[/tex] are angles, [tex]\varphi[/tex] is my parameter, [tex]\theta[/tex] and [tex]\beta[/tex] are some variable angles.

    I need to find the coordinates of the intersection of curve and the ray as a function of [tex]\theta[/tex] and [tex]\beta[/tex].

    In short I want to know coordinates of intersection but without the parameter [tex]\varphi[/tex] in them.
     
  2. jcsd
  3. Sep 8, 2010 #2
    Mach18Image1.png

    This is the picture of my problem for r = 5, R= 6 and D = 7.
    Note that [tex]\theta[/tex] is not visible on this picture, but it had a fixed value while I was taking the picture.
    Making [tex]\theta[/tex] change would cause curve to change position and orientation.

    Thanks :D
     
  4. Sep 8, 2010 #3
    How about numerically? It's not too hard to design an algorithm that zeros into the roots without having to manually select starting values.
     
  5. Sep 9, 2010 #4
    Hmmm. I need distance of that intersection from the origin of the coordinate system. But I need it as a function of a [tex]\theta[/tex] so I can find its minimal value for different [tex]\theta[/tex]s . (i would have to use derivatives and other tricks later)
     
  6. Sep 9, 2010 #5
    Well what, that ain't no hill neither. But first you need to make clear if you are wiling to settle for a numeric approximation. If not, then I can't do it.
     
    Last edited: Sep 9, 2010
  7. Sep 10, 2010 #6
    I am sorry, but numeric solution doesn't help me, since I cannot analyze the data further that way, and I also don't think it's an easy task.

    Maybe I am using a wrong approach.


    Here more description of my problem:
    I have to change [tex]\theta[/tex] from 0 to some value. While changing it my (green) curve moves (the origin of the coordinate system is always inside of it).
    If I trace my curve, I get a unusual white shaped area. I want to find the equation for the boundary line of that area.

    mach18image2.png
     
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