Drawing Curve in R2 Space: ds2=y2dx2+dy2

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SUMMARY

The discussion focuses on drawing a curve in R2 space defined by the line element ds2=y2dx2+dy2. Participants clarify that this line element does not directly correspond to a curve in Euclidean R2, which is governed by the induced metric ds = √(1+(df/dx)²)dx. The concept of an enveloping space is introduced, with a specific mapping phi(x,y)=(x/y,√(x²+y²)) being proposed as a potential enveloping space for the curve. The term "enveloping space" is questioned, indicating a need for further clarification on its definition.

PREREQUISITES
  • Understanding of R2 space and Euclidean metrics
  • Familiarity with differential geometry concepts
  • Knowledge of line elements and induced metrics
  • Basic understanding of mappings and transformations in mathematics
NEXT STEPS
  • Research the concept of enveloping spaces in differential geometry
  • Study the implications of line elements in R2 space
  • Learn about the induced metric from Euclidean geometry
  • Explore transformations such as x'=arctan(y/x) and y'=√(x²+y²)
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Mathematicians, physicists, and students studying differential geometry or interested in the properties of curves in R2 space.

maddy
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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?

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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Originally posted by maddy
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
<br /> ds = \sqrt{1+\left(\frac{df}{dx}\right)^2}dx<br />
from the euclidean metric
<br /> ds^2 = dx^2 + dy^2<br />

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?
 

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