How to obtain a 2D-coordinate system from two family of curves?

In summary, it is possible to determine a coordinate system from two families of curves if you know the equations that define the curves.
  • #1
mnb96
715
5
Hello,

it is known that if we have a curvilinear coordinate system in ℝ2 like [itex]x=x(u,v)[/itex], [itex]y=y(u,v)[/itex], and we keep one coordinate fixed, say [itex]v=\lambda [/itex], we obtain a family of one-dimensional curves [itex]C_{\lambda}(u)=\left( x(u,\lambda),y(u,\lambda) \right)[/itex]. The analogous argument holds for the other coordinate u. These family of curves are sometimes called coordinate lines, or level curves.

My question is: if I am given two family of curves [itex]C_v(u)[/itex] and [itex]C_u(v)[/itex] is it possible to obtain the system of curvilinear coordinates [itex]x(u,v)[/itex], [itex]y(u,v)[/itex] that generated them?
 
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  • #2
Consider the family of radial curves and a family of nested circles. Together they create polar coordinates.

Can you determine the curvilinear coordinate system from that knowledge?
 
  • #3
Yes, if we have a family of circles [itex]C_r(\theta)=\left( r\cos\theta, r\sin\theta \right)[/itex] for some [itex]r\in \mathbb{R}^+[/itex], and a family of straight lines passing through the origin [itex]C_\theta(r)=\left( r\cos\theta, r\sin\theta \right)[/itex] for some [itex]\theta\in[0,2\pi)[/itex] the solution is quite trivial.

I was interested more in a general procedure or simply a strategy that I could follow to solve this kind of problem.

If we cannot answer the general question then let's try at least a less trivial example I was unable to solve like this one: we have two families of "parallel" exponential curves, the first family is [itex]C_\lambda(u) = (u, \; e^u +\lambda)[/itex] for some fixed real scalars v, and the other family is [itex]C_k(v) = (e^{-v} + k, \; v)[/itex] for some real k.
I was unable to obtain two functions x(u,v) , y(u,v) such that [itex]C_\lambda(u) = (x(u,\lambda), \; y(u,\lambda))[/itex] and [itex]C_k(v)=(x(k,v),\; y(k,v))[/itex]
 
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  • #4
so i guess a way to investigate this is to determine if you lost any info when generating the two sets of curves such that you would find multiple different answers when you reverse the problem.
 
  • #5
well I imagine you'd have to check the curves, not all curves form coordinates systems.
 

1. How do I determine the origin of the 2D-coordinate system?

The origin of the 2D-coordinate system is the point where the two family of curves intersect. This can be found by setting the equations of the two curves equal to each other and solving for the coordinates of the point of intersection.

2. How do I label the x and y axes of the 2D-coordinate system?

The x and y axes are typically labeled based on the variables used in the equations of the two family of curves. For example, if the equations use x and y as variables, then the x-axis would be labeled as x and the y-axis as y.

3. Can I use any two family of curves to create a 2D-coordinate system?

Yes, any two family of curves can be used to create a 2D-coordinate system. However, it is important to ensure that the curves intersect and that the resulting coordinate system accurately represents the data being studied.

4. How do I plot points on a 2D-coordinate system from the equations of the two curves?

To plot points on a 2D-coordinate system, you can substitute values for the variables in the equations of the two curves and solve for the corresponding coordinates. These coordinates can then be plotted on the coordinate system to create a visual representation of the data.

5. How can I use a 2D-coordinate system to analyze data?

A 2D-coordinate system can be used to analyze data by plotting points and identifying patterns or trends in the data. It can also be used to make predictions or draw conclusions based on the behavior of the curves in the coordinate system.

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