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Math Help - conformal mapping (help needed)

  1. #1
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    conformal mapping (help needed)

    Im mechanics engineering student and Im working on torsion problem of elasticity. I would be thankful if someone teaches me how to map a non-circular closed curve with components of:

    x=L*(cos(theta)+W*cos(n*theta))
    y=L*(sin(theta)-W*sin(n*theta))

    onto a circle. ( L resembles radius, n and W control the shape of the closed curve)
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  2. #2
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    So you want a transformation from (x,y) to (u,v) s.t u and v represent a circle, you might as well write it down as:
    u=Ax+By
    v=Cx+Dy
    u^2+v^2=(A^2+C^2)x^2+(B^2+D^2)y^2+2(DC+AB)xy
    xy=L^2(cos(theta)sin(theta)-W^2cos(n*theta)sin(n*theta))
    x^2=L^2(cos^2(theta)+2Wcos(theta)cos(n*theta)+W^2c os^2(n*theta)
    y^2=L^2(sin^2(theta)-2Wsin(thta)sin(n*theta)+W^2*sin^2(n*theta))
    Set A^2+C^2=B^2+D^2=1
    sowe get: x^2+y^2=L^2(1+W^2-2Wcos((n+1)*theta))

    Now also DC+AB=1.

    There are multiple choices to choose from, but watch also for the fact that W and theta are dependent on eachother, to find this dependence, you need to eqaute u^2+v^2=L^2.

    Now ofcourse my approach is valid if W and theta are indeed dependent in the question, if not then you might not get a radius L from this transformation.
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