Drawing Curve in R2 Space: ds2=y2dx2+dy2

  • Thread starter maddy
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In summary, the conversation discusses the possibility of drawing a curve with the line element ds2=y2dx2+dy2 in a R2 space. There is confusion over how this curve can be mapped onto the enveloping space phi(x,y)=(x/y, sqrt(x2+y2)). There is also a discussion about the concept of an enveloping space and its relation to the original Euclidean space.
  • #1
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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  • #2
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
[tex]
ds = \sqrt{1+\left(\frac{df}{dx}\right)^2}dx
[/tex]
from the euclidean metric
[tex]
ds^2 = dx^2 + dy^2
[/tex]

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?
 
  • #3
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?
 

What is "Drawing Curve in R2 Space"?

"Drawing Curve in R2 Space" refers to the process of graphing a curve on a two-dimensional coordinate plane using the equation ds2=y2dx2+dy2. This equation represents a specific type of curve known as a parametric curve, where the x and y coordinates are represented by separate equations in terms of a third parameter, typically denoted by t.

What is R2 Space?

R2 Space, also known as two-dimensional Euclidean space, is a mathematical concept that represents a two-dimensional coordinate plane with orthogonal axes x and y. In this space, points are represented by their x and y coordinates, and curves can be graphed using equations that relate these coordinates.

What does the equation ds2=y2dx2+dy2 represent?

The equation ds2=y2dx2+dy2 represents a parametric curve in R2 Space, where the x and y coordinates are defined by separate equations in terms of a third parameter, typically denoted by t. This type of curve can have complex shapes and can be used to represent various mathematical functions or real-world phenomena.

How do you graph a curve in R2 Space using the equation ds2=y2dx2+dy2?

To graph a curve in R2 Space using the equation ds2=y2dx2+dy2, you can follow these steps:

  1. Choose a range of values for the parameter t, typically from 0 to 1.
  2. Substitute each value of t into the equations for x and y to calculate the corresponding x and y coordinates.
  3. Plot these points on the coordinate plane.
  4. Connect the points with a smooth curve to visualize the shape of the curve.

You can also use a graphing calculator or software to plot the curve automatically.

What are some examples of real-world applications of "Drawing Curve in R2 Space"?

"Drawing Curve in R2 Space" has various applications in fields such as physics, engineering, computer graphics, and data visualization. Some examples include representing the trajectory of a projectile, visualizing the motion of a particle in a magnetic field, creating 2D animations, and plotting complex mathematical functions.

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