Can y^2 = x^2 be parametrized?

[Unit]

The graph of y^2 = x^2 (1) looks simply like y = x (2) and y = -x (3) plotted on the same axis.

Is it possible to parameterize (1)?

 

[Werg22]

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.

 

[Unit]

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"?

 

[mathman]

This may be silly, but:
x²=t and y²=t

 

[Unit]

This may be silly, but:
x²=t and y²=t

Hahaha, I knew that . I meant, x = some function of t and y = some function of t. And I don't think x = \pm \sqrt{t} and y = \pm \sqrt{t} counts.

 

[Werg22]

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"?

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.