Inverting parabolic and stereographic coordinates

In summary, the given parabolic co-ordinate system can be transformed into Cartesian coordinates x and y using the equations x=\pm \sqrt{\frac{\lambda\pm\sqrt{\lambda^2+1}}2} and y=\frac{2\mu}{\pm \sqrt{\frac{\lambda\pm\sqrt{\lambda^2+1}}2}}. However, to find the inverse transformation for general "nonlinear" co-ordinate systems, such as the stereographic projection defined by \frac{x}{\xi} = \frac{y}{\eta} = \frac1{1-\zeta}, it may be necessary to substitute variables and solve for the desired variables.
  • #1
Epimetheus
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Homework Statement



Given the parabolic co-ordinate system defined, given Cartesian coordinates x and y, as
[tex]$\mu=2xy$[/tex]
[tex]$\lambda = x^2-y^2,$[/tex]
find the inverse transformation [tex]x(\mu, \lambda)[/tex] and [tex]y(\mu,\lambda)[/tex].

Homework Equations



None

The Attempt at a Solution



We compute [tex]\lambda/\mu^2[/tex] in order to obtain
[tex]x=\pm \sqrt{\frac{\lambda\pm\sqrt{\lambda^2+1}}2}[/tex]
and
[tex]y=\frac{2\mu}{\pm \sqrt{\frac{\lambda\pm\sqrt{\lambda^2+1}}2}}[/tex]

However, I don't think this is right ... I'm following a set of CM lecture notes, which immediately go on to claim [tex]x^2+y^2=\lambda^2+\mu^2[/tex] and
[tex]\dot x^2 + \dot y^2 = \frac14 \frac{\dot \lambda^2 + \dot \mu^2}{\sqrt{\lambda^2 + \mu^2}} [/tex]
which looks tantalizingly similar to spherical polars, but doesn't seem to follow from what I have ...

Any thoughts on inverting general "nonlinear" co-ordinate systems (other than cylindrical and spherical)? What about finding the inverse transformation for the stereographic projection defined by
[tex]\frac{x}{\xi} = \frac{y}{\eta} = \frac1{1-\zeta}[/tex]

so that [tex]x=\frac{\xi}{1-\zeta}; y=\frac{\eta}{1-\zeta} [/tex]?

It's hard to get rid of the "old" set of variables in this case.
 
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  • #2
If you want to find x and y in terms of [tex]\lambda[/tex] and [tex]\mu[/tex], then youre given [tex]\mu=2xy[/tex]

[tex]y=\frac{\mu}{2x}[/tex].

Substitute this in the other equation and solve for x. Similarly for y.
 
Last edited:
  • #3


As a scientist, it is important to first clarify what is meant by "inverting" a coordinate system. In general, this refers to finding the inverse transformation that maps points from the new coordinate system back to the original Cartesian coordinates. In the case of the parabolic coordinate system given, the inverse transformation can be found by solving the equations for x and y in terms of $\mu$ and $\lambda$. However, it is important to note that this is not a unique solution, as there are multiple ways to express x and y in terms of $\mu$ and $\lambda$. It is also important to check the validity of the inverse transformation, such as ensuring that the Jacobian determinant is non-zero.

In terms of finding the inverse transformation for other "nonlinear" coordinate systems, it can be a challenging task and may require advanced mathematical techniques. It is important to carefully consider the relationships between the new coordinates and the original Cartesian coordinates in order to find the appropriate inverse transformation. The stereographic projection you have provided is a good example of this, as it involves a transformation from 3D Cartesian coordinates to 2D coordinates on a plane. In this case, it may be helpful to consider the geometric interpretation of the stereographic projection and use trigonometry to find the inverse transformation.

In summary, inverting coordinate systems requires careful consideration and may involve advanced mathematical techniques. It is important to ensure the validity of the inverse transformation and to carefully interpret the relationships between the new coordinates and the original Cartesian coordinates.
 

Related to Inverting parabolic and stereographic coordinates

What are parabolic and stereographic coordinates?

Parabolic coordinates are a coordinate system that describes points in a plane using two parameters, typically denoted as u and v. Stereographic coordinates, on the other hand, are a coordinate system that maps points on a sphere onto a plane using a single parameter, typically denoted as θ. These coordinate systems are widely used in mathematics and physics, particularly in the study of conic sections and spherical geometry.

How are parabolic and stereographic coordinates related?

Parabolic and stereographic coordinates are related through a geometric transformation known as inversion. Inversion is a type of transformation that maps points in one coordinate system onto points in another coordinate system through a specific mathematical formula. In the case of parabolic and stereographic coordinates, the inversion formula is given by u = 2x/(x²+y²), v = 2y/(x²+y²), and θ = 2arctan(y/x).

What is the purpose of inverting parabolic and stereographic coordinates?

The inversion of parabolic and stereographic coordinates allows for the conversion of points between these two coordinate systems. This is useful in many applications, such as in solving mathematical problems involving conic sections and spherical geometry, as well as in mapping the Earth's surface onto a two-dimensional plane.

What are the advantages of using parabolic and stereographic coordinates?

One of the main advantages of using parabolic and stereographic coordinates is that they provide a more natural and intuitive way of representing points in a plane and on a sphere, respectively. They also have many useful mathematical properties, such as being conformal and preserving angles and shapes.

Are there any limitations or challenges in using parabolic and stereographic coordinates?

One limitation of using parabolic and stereographic coordinates is that they are not suitable for all types of problems. They are most commonly used in situations where the shape or geometry of the problem can be described by a parabola or a sphere. Additionally, the inversion formula can be quite complex and may require advanced mathematical knowledge to understand and use effectively.

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