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Curve with ever increasing radius

  1. May 16, 2007 #1
    Cannot remember the name of a curve that begins with a lesser radius and tangents off into an ever increasing radius until it is almost a straight line.

    Very commonly used in art deco design.

    Thanks for your help.
     
    Last edited: May 16, 2007
  2. jcsd
  3. May 16, 2007 #2
    Come on, some one has to know the name

    You see it on the more complex mechanical drawing templates, which I don't have.
     
  4. May 16, 2007 #3

    robphy

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    Are you thinking of a spiral?
     
  5. May 16, 2007 #4
    No, the curve I want can be on a plane(one dimension)

    or not and this site won't let me draw it to show it to you. It starts like a spiral with a tighter curve but the second curve swings open eventually to almost a straight line as it progresses to infinity(becoming an nearly imperceptible curve as it progresses away from the first curve due its much larger and ever increasing radius). Perhaps I should ask the physics folks as this curve is commonly seen in astronomy.
     
  6. May 16, 2007 #5

    robphy

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  7. May 17, 2007 #6

    Chris Hillman

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    Which spiral?

    He might mean the "hyperbolic spiral" http://mathworld.wolfram.com/HyperbolicSpiral.html which in polar coordinates has the equation [itex]r \, \theta = a[/itex], and which is asymptotic to [itex]y=a[/itex]. But if so, "begins with a lesser radius" doesn't sound right.

    The more familiar logarithmic spiral http://mathworld.wolfram.com/LogarithmicSpiral.html [itex]r = \exp(a \, \theta)[/itex] has no such asymptote, and has the property that the curve intersects each ray infinitely often but makes the same angle each time it intersects a given ray.

    As for "commonly seen in astronomy", I guess he might mean the "parabolic spiral" http://mathworld.wolfram.com/FermatsSpiral.html [itex](r-a)^2 = 4 a \, k \, \theta[/itex], which to some eyes vaguely resembles the arm of a spiral galaxy (but physicists know that these "arms" are to some extent optical illusions).

    Finally, it is possible he is confusing the clothoid or "Euler-Cornu spiral" http://mathworld.wolfram.com/CornuSpiral.html with the hyperbolic spiral.
     
    Last edited: May 17, 2007
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