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Monotone Test of the equation

  1. May 15, 2009 #1
    Hi all:

    Assume in 3D space there is a point [itex]v=[v_x, v_y, v_z][/tex], and a normal vector associate with it as [tex]n=[n_x, n_y, n_z][/tex]. A line function is defined as [tex]u=w+t\cdot l[/tex] where [tex]w=[w_x, w_y, w_z][/tex] is a point, and [tex]l=[l_x, l_y, l_z][/tex] is the normalized direction of the line. l and n are normalized. Assume there is a function defined as:

    [tex]
    K = \frac{(u-v)\cdot n}{||u-v||^2}
    [/tex]

    My question is when point u varies on the line, is the function K varies monotonically???

    I've tried to compute [tex]\frac{dK}{dt}[/tex], but I can't really see if it's monotone or not, can some one help me please?

    Thanks
     
    Last edited: May 15, 2009
  2. jcsd
  3. May 15, 2009 #2

    tiny-tim

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    Science Advisor
    Homework Helper

    Hi Asuralm! :smile:

    Forget calculus, this is geometry :wink:

    Hunt: if u and v represent points U and V, and if the nearest point on L to U is N, what is (u - v).n ? :smile:
     
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