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Logarithmic Spirals

  1. Mar 23, 2013 #1
    1. The problem statement, all variables and given/known data

    This is a logarithmic spiral. How can I find the length of the spiral (curve) from the origin to point X on the curve? The distance between the origin and the point is given (we can call that the radius) and the distance between the origin and point A is given which this is the same between points A and B and C. Point X is on the spiral with an indicated arrow pointing towards it. The location, therefore, the angle between the origin and this point is unknown. However the distance is known. How can I find the total length of the spiral from the origin to point X?

    Mod note: The text above is from the oversized image originally posted. A cropped version of the image is below.

    attachment.php?attachmentid=57047&stc=1&d=1364122742.jpg

    2. Relevant equations

    1. ##r=ae^{b\theta}##
    2. ##L=r\theta##


    3. The attempt at a solution

    Do I need to solve for theta then use L=r(theta) to find the total length?
    I'm having troubles finding the terms for the first equation.
     

    Attached Files:

    Last edited by a moderator: Mar 24, 2013
  2. jcsd
  3. Mar 23, 2013 #2

    mfb

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    That is a good idea.
    You know that A and B satisfy the equation for the spiral. This allows to determine a and b.
     
  4. Mar 23, 2013 #3
    Are you referring to the points A and B? So my values at point A and B correspond to the a and b terms in the equation?
     
  5. Mar 24, 2013 #4

    mfb

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    Staff: Mentor

    Sure
    No. The names are a bit misleading in that respect.
     
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