# Y^2 = x^2

1. May 18, 2010

### 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)?

2. May 19, 2010

### 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.

3. May 19, 2010

### 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"?

4. May 19, 2010

### mathman

This may be silly, but:
x2=t and y2=t

5. May 19, 2010

### Unit

Hahaha, I knew that :tongue:. 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.

6. May 19, 2010

### Werg22

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.