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Closest line from a point to a curve in R^2

  1. Jan 19, 2016 #1

    RBG

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    Given a parametrized curve ##X(t):I\to\mathbb{R}^2## I am trying to show given a fixed point ##p##, and the closest point on ##X## to ##p##, ##X(t_0)##, the line between the point and the curve is perpendicular to the curve. My only idea so far is to show that ##(p-X(t))\cdot(\frac{X'(t)}{||X'(t)||})=0##. But in general, I don't see why this would be true? It seems clear geometrically, but obviously that's not an argument. Any hints?
     
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  3. Jan 19, 2016 #2

    Simon Bridge

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    Use the definition of "perpendicular".
     
  4. Jan 19, 2016 #3

    RBG

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    Isn't that the dot product is zero? That or that the slopes of the tangent lines are inverse reciprocals of one another. But I don't see how the latter definition can be applied...
     
  5. Jan 19, 2016 #4

    Simon Bridge

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    Dot products of what?
    How can a line be perpendicular to a curve?
     
  6. Jan 19, 2016 #5

    Krylov

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    I take it that you assume the curve is smooth. Even then, note that the closest point may not be unique. Also, when ##I## is not compact, a closest point is not guaranteed to exist, as ##t \mapsto \|p - X(t)\|## need not assume its infimum over ##I## then.
     
  7. Jan 19, 2016 #6

    RBG

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    Dot product of the tangent vector, right? So above ##(p-X(t_0))## is the vector between point and curve and ##X'(t_0)## is the tangent vector. But I don't see why should ##p\dot X'(t_0)-X(t_0)X'(t_0)=0##
     
  8. Jan 19, 2016 #7

    Simon Bridge

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    ... given a point on the curve, how would you tell that it is the closest point?
    Given the curve C and a point P, how would you usually go about finding the closest point on C to P?
     
  9. Jan 19, 2016 #8

    RBG

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    We are assuming ##X(t)## is a regular parametrized curve and ##t_0## is not an endpoint of ##I##.
     
  10. Jan 19, 2016 #9

    Simon Bridge

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    ... why not just work through the expression that arises from the definition you just used?
     
  11. Jan 19, 2016 #10

    RBG

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    You would take the derivative of ##||p-X(t)||## and minimize it. Then check which points are minimal, right?
     
  12. Jan 19, 2016 #11

    Ray Vickson

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    That is how I would do it, but I would make the problem easier by minimizing ##|| p - X(t)||^2 ## instead of ##||p - X(t)||##. These problems are equivalent, in the sense that their ##t##-solutions are the same.
     
  13. Jan 19, 2016 #12

    RBG

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    I really don't understand what you mean by this. I should use the fact that I am minimizing ##|p-X(t)||## somehow to reduce the dot product?
     
  14. Jan 19, 2016 #13

    RBG

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    OOOOOOHHHHH... duh. Nevermind. Right... Thanks! Just do the calculation of taking the derivative
     
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