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A Curve under conformal mapping

  1. Jul 23, 2016 #1
    I know the concepts of conformal mapping and complex mapping but I didn’t see none explanation about how apply this ideia and formula for convert a curve, or a function, between different maps.

    Look this illustration…

    Duvida.png
    In the Cartesian map, I basically drew a liner function f(x) = ax+b (defined by part). Which would be the graph curve and the algebraic expression equivalent in the polar map and log-polar map?
     
  2. jcsd
  3. Jul 23, 2016 #2

    fresh_42

    Staff: Mentor

    Why don't you define ##f## properly and simply transform the coordinates?

    ##(x,f(x)) = (\sqrt{x^2+f(x)^2}, \arccos \frac{x}{\sqrt{x^2+f(x)^2}}) = (\log \sqrt{x^2+f(x)^2}, \arccos \frac{x}{\sqrt{x^2+f(x)^2}})##
     
  4. Jul 23, 2016 #3
    Because I want (actually, I need, due the technical difficulties) to express ##r = r(\theta)##

    EDIT: I can't to express an implicit funcion in polar or log polar mode....
     
  5. Jul 23, 2016 #4

    fresh_42

    Staff: Mentor

    Radius and angle are related by ##\cos θ = \frac{x}{r}## or ##\sin θ = \frac{f(x)}{r}## and ##r = \sqrt{x^2 + f(x)^2}.## Of course you will have to keep an eye on signs, resp. the range of the angle. In the logarithmic version in my understanding only the radius will be affected, i.e. the coordinate becomes ##\log (r)##
    Maybe I didn't get what you meant.
     
  6. Jul 23, 2016 #5

    fresh_42

    Staff: Mentor

    You don't express functions in "polar mode", implicit or not. You can only express points in coordinates.
    Therefore you write ##(x,y) = (x,f(x))## or ##(x,y) = (r(x,y), θ(x,y))##. A function expresses how a value ##y = f(x)## varies if ##x## does. The ##x-##axis is already an interpretation and a graph ##(x,f(x))## a visualization.
     
  7. Jul 23, 2016 #6
    ##r = \sqrt{x^2 + y^2}## ......... ##(y = ax+b)##

    ##r = \sqrt{x^2 + (a x + b)^2}## ......... ##(x = r \cos(\theta))##

    ##r = \sqrt{(r \cos(\theta))^2 + (a (r \cos(\theta)) + b)^2}##

    https://www.wolframalpha.com/input/?i=r+=+sqrt((r+cos(t))²+++(a+r+cos(t)+++b)²)+solve+for+r

    "r = sqrt((r cos(t))² + (a r cos(t) + b)²) solve for r"

    solutions:

    ##r = - \frac{b}{a \cos(t) - \sin(t)}##

    ##r = - \frac{b}{a \cos(t) + \sin(t)}##

    ploting:

    final.png

    So...

    This answer is, of course, too much interesting!!! But, it doesn't the answer that I'm looking for...

    I want that the straight of the Cartesian map becomes a curve in the polar and log polar mapping.
     
    Last edited: Jul 23, 2016
  8. Jul 24, 2016 #7
    No more answers??
     
  9. Jul 25, 2016 #8

    FactChecker

    User Avatar
    Science Advisor
    Gold Member

    In general, that can not be done. There can be many r values associated with the same ##\theta## value. In the example you give, if the origin is at the center point of the first diagram, then there are entire path segments with the same ##\theta## value.

    Paths should be parameterized using a separate parameter, t, in [0,1].
     
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