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Y^2 = x^2

  1. May 18, 2010 #1
    The graph of [itex]y^2 = x^2[/itex] (1) looks simply like [itex]y = x[/itex] (2) and [itex]y = -x[/itex] (3) plotted on the same axis.

    Is it possible to parameterize (1)?
  2. jcsd
  3. May 19, 2010 #2
    Parametrized by a continuous curve? If so, no, because removing the origin from y^2 = x^2 gives 4 pieces, whereas removing a point from an interval (a, b) gives 2 pieces. More precisely, the image of (a,c) U (c, b) where a < c < b is going to be the union of two connected sets, and this can't possibly be 4 non-intersecting connected pieces.

    If not continuous, then yes, simply by a cardinality argument.
  4. May 19, 2010 #3
    Thanks Werg22! I meant parametrized by expressing y in terms of t and x in terms of t. I understand why it can't be a continuous curve. How would you parametrize it, not continuously, "by a cardinality argument"?
  5. May 19, 2010 #4


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    This may be silly, but:
    x2=t and y2=t
  6. May 19, 2010 #5
    Hahaha, I knew that :tongue:. I meant, x = some function of t and y = some function of t. And I don't think [itex]x = \pm \sqrt{t}[/itex] and [itex]y = \pm \sqrt{t}[/itex] counts.
  7. May 19, 2010 #6
    I don't know of any explicit parametrization, but any proper real interval has the same cardinality as y^2 = x^2, so an onto map from (a,b) to y^2 = x^2 exists. This is simply an existential statement, take it for what it's worth.
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