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How many equations are there for the unit circle?

  1. Nov 18, 2009 #1
    hopefully we all know x^2 + y^2 =1 and x=cost y=sint, t between 0 and 2pi.

    There's also one with slope;

    x= (1-t^2)/(1+t^2)
    y= (2t)/(1+t^2)

    I was wondering if this counts as a separate one

    x+iy=e^it, t also between 0 and 2pi

    or if this is analogous to the trig parameterization. I don't know a whole lot of trig(I'm not really a math man per se), but something in my gut tells me these two aren't really different... Anyway, I was just curious. I'd also be interested in any other ones.

    I don't know much about hyperbolic geometry, can there be a unit circle for that?

    I'm also starting to work through something as a set of ratios, but does anyone else have any other ideas?
     
  2. jcsd
  3. Nov 18, 2009 #2

    mathman

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    It is analogous. Euler's identity eit = cost + isint.
     
  4. Nov 18, 2009 #3
    There's also r=cost and r=sint, i don't know if you'd count them as unit circles though (they are off centre).
     
  5. Nov 19, 2009 #4
    How about x^2 + y^2 = 1 + a - a for any a? :tongue:

    It doesn't make much sense to ask this question, there are infinitely many equations we can come up with.
     
  6. Nov 19, 2009 #5
    This derives from the first by setting
    [itex]
    t=\tan\frac{a}{2}
    [/itex]
    An in fact they all can be found this way. Plug in any function (that has the required range) for a variable.
     
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