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Circles and Euler spiral (repost from general math)

  1. May 5, 2017 #1
    Hi,
    i have this problem..., giving two circle (example radius 1 = 500 units, radius 2 = 200 units, distance between centers = 275.73 units) find Euler Spiral (aka Cornu spiral, aka Clothoid https://en.wikipedia.org/wiki/Euler_spiral) tangent giving circle (unknown tangent points).
    For this problem are two mirrored Euler spiral as solution with lenght = 450 units.
    Problem so simply to explain but not so simply to find solution......
    Many thanks for help......
    XjDbYHOr_Br-kCIA9dbKtPZ3_hF3oLd2wvA82IBgzcQ?dl=0&size=1280x960&size_mode=3.png
     
  2. jcsd
  3. May 5, 2017 #2

    BvU

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    Science Advisor
    Homework Helper
    Gold Member

    Hello again, Mario, and :welcome: again, too

    Well, we are making progress. There is a problem statement and the link gives a bunch of equations. So we can claim 1 and 2 below are filled in.
    All we have to do now is make a start with 3 !

    1. The problem statement, all variables and given/known data


    2. Relevant equations


    3. The attempt at a solution

    Oh, and: don't erase the template (or was your thread moved here by a moderator?)

    For a start: how did you find the 275.73 ?
     
  4. May 6, 2017 #3
    1. The problem statement, all variables and given/known data

    known data:
    center of circle 1 (xO1,yO1) and radius of circle = R1
    center of circle 2 (xO2,yO2) and radius of circle = R2

    unknown data:
    lenght of spiral (magenta and blue) L=?
    tangent point of spiral on circle 1 (xA, yA) = ?
    tangent point of spiral on circle 2 (xB, yB) = ?

    2. Relevant equations


    Particular declination of problem are resolved with below equation, but in this resolution know data are:
    R1, R2, L, xA=zero, yA=zero, xO1=zero, yO1=R1, line(O1-A) is orthogonal to x axes
    calculated data are:
    xO2, yO2, xB, yB
    E0SKIm4P2938lm3JMCH_vwOsLQWlWuyP7WGDQ5v-mJg?dl=0&size=2048x1536&size_mode=3.png
    RWiXdEsMMiiWUQ1ILFSpS2yKHG1QW3iBUXhgVIgtWOc?dl=0&size=2048x1536&size_mode=3.png
    k7ROPOg-Qfh8T1uy89Tmp0PlB_lKnOLIdIQ1rLqi_z0?dl=0&size=1600x1200&size_mode=3.png
    HdVyqmRSCaP-oZm8RR8FCKFmWVrehBDMS5t_4caiz1s?dl=0&size=1600x1200&size_mode=3.png
    (complete document at this link http://www.mygeodesy.id.au/documents/Horizontal Curves.pdf prof. R.E.Deakin)

    3. The attempt at a solution

    A iterative solution is using the above equations and change L to find desidered distance of two circle...
    but Analitical solution is possible ?
    many thanks

    P.S. move post here, if I understood well, it was a hint
     
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