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Visual complex analysis problem

  1. Jul 22, 2011 #1
    1. The problem statement, all variables and given/known data

    Explain geometrically why the locus of z such that

    arg [ (z-a)/(z-b) ] = constant

    is an arc of a certain circle passing through the fixed points a and b.


    i tried to visualize the equation in a cartesian co-system but in doing so, i was not very successful.
     
  2. jcsd
  3. Jul 22, 2011 #2

    tiny-tim

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    hi raphael3d! :wink:
    no, visualise these problems as a diagram in Euclidean geometry, not as an equation …

    what do you get? :smile:
     
  4. Jul 22, 2011 #3
    an ellipse, is my guess?
     
  5. Jul 22, 2011 #4

    tiny-tim

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    no, i mean describe what "arg [ (z-a)/(z-b) ] = constant" means as a piece of geometry …

    what lines is it telling you to draw? :wink:
     
  6. Jul 22, 2011 #5
    two lines from two distinctive points a,b to one point z. whereas those lines form angles with the horizontal and the difference between those angles is constant. all the points lie on a circle...
    i have drawn the lines and points and angles, but i dont know how to proceed from here... what kind of circle and so forth...
     
  7. Jul 22, 2011 #6

    tiny-tim

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    hmm …

    a better way of putting it is that from two points a and b, we draw a pair of lines that meet at a given angle

    you should be able to prove that all such points (for a fixed angle) form an arc of a circle :wink:
     
  8. Jul 23, 2011 #7
    if a is 1 and b is i on the unit circle, then z lies in the first quadrant? i would guess the angle where a and b meet z doesnt change as long as z lies between them...?
     
  9. Jul 23, 2011 #8
    you mean...meet at a given angle c?

    i am stuck, to be honest^^
     
    Last edited: Jul 23, 2011
  10. Jul 23, 2011 #9

    tiny-tim

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    hi raphael3d! :smile:

    (just got up :zzz: …)

    there's a well-known theorem that the locus of points which subtend a fixed angle from two given points is an arc of a circle joining those two points :smile:

    you need to find a book of geometry (sorry, i don't know any online ones :redface:) which gives you all the theorems for a circle, and their proofs …

    clearly this is background knowledge which your course assumes you already have​
     
  11. Jul 23, 2011 #10
    clearly we live in shifted time zones =)

    well thank you, i will look into that...surely there will be a wiki or something similar.

    this is a problem of "visual complex analysis" by tristan needham. a wonderful book :)

    here is it:
    http://www.mathsisfun.com/geometry/circle-theorems.html

    now i would love to show it with some complex algebra ;)

    thanks for the help
    keep up the good work, with that many qualitative posts you could easily have written a book.

    metta
     
  12. Jul 23, 2011 #11

    tiny-tim

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    hi metta! :smile:
    yes, that looks good

    the theorem you need is the third diagram on that page, marked "Angles Subtended by Same Arc Theorem" :wink:
     
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