# Enveloping spaces

How do you draw a curve with line element ds2=y2dx2+dy2 in a R2 space? Is it just lots of x=a lines, with a-any real number?

I don't understand why a mapping of phi(x,y)=(x/y,sqrt(x2+y2)) can be an enveloping space for the above curve?

Any ideas, anyone?

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How do you draw a curve with line element ds2=y2dx2+dy2 in a R2 space? Is it just lots of x=a lines, with a-any real number?
this doesn t quite make sense. if you draw a curve in R2, it inherits an induced metric from the normal euclidean metric.

this metric is
$$ds = \sqrt{1+\left(\frac{df}{dx}\right)^2}dx$$
from the euclidean metric
$$ds^2 = dx^2 + dy^2$$

i cannot get the line element you wrote from a curve in euclidean R2
I don't understand why a mapping of phi(x,y)=(x/y,sqrt(x2+y2)) can be an enveloping space for the above curve?

Any ideas, anyone?
what is an enveloping space? i have never heard this term before. can you define it please?

what is an enveloping space? i have never heard this term before. can you define it please?
It's from this paper gr-qc/9405063.

this doesn't quite make sense. if you draw a curve in R2, it inherits an induced metric from the normal euclidean metric.
Can I substitute make x'=arctan(y/x) and y'=sqrt(x2+y2) into ds2=dx2+dy2 so that phi'(x,y)=(x',y') be the covering space for the original Euclidean space with (0,0) removed?