Parabolic Coordinates Radius

In summary, the conversation discusses the relationship between Cartesian, Spherical, and Parabolic coordinate systems. The speaker asks for clarification on the orientation of the coordinate system and why the radius in x and y is represented by the term \sqrt{ \varepsilon \eta }. The other person explains that the product of \varepsilon and \eta equals the height, which is represented by the radius in the x-y plane. This relationship is mathematically shown as x = \sqrt{ \varepsilon \eta } \cos (\phi) and y = \sqrt{ \varepsilon \eta } \sin (\phi). However, the speaker still lacks physical or geometrical intuition for this relationship.
  • #1
bolbteppa
309
41
Given Cartesian [itex](x,y,z)[/itex], Spherical [itex](r,\theta,\phi)[/itex] and parabolic [itex](\varepsilon , \eta , \phi )[/itex], where

[tex]\varepsilon = r + z = r(1 + \cos(\theta)) \\\eta = r - z = r(1 - \cos( \theta ) ) \\ \phi = \phi [/tex]

why is it obvious, looking at the pictures

WZ0CY.png


ccUqO.png


(Is my picture right or is it backwards/upside-down?)

that [itex]x[/itex] and [itex]y[/itex] contain a term of the form [itex]\sqrt{ \varepsilon \eta }[/itex] as the radius in

[tex]x = \sqrt{ \varepsilon \eta } \cos (\phi) \\ y = \sqrt{ \varepsilon \eta } \sin (\phi) \\ z = \frac{\varepsilon \ - \eta}{2}[/tex]

I know that [itex] \varepsilon \eta = r^2 - r^2 \cos^2(\phi) = r^2 \sin^2(\phi) = \rho^2[/itex] ([itex]\rho[/itex] the diagonal in the x-y plane) implies [itex] x = \rho \cos(\phi) = \sqrt{ \varepsilon \eta } \cos (\phi) [/itex] mathematically, but looking at the picture I have no physical or geometrical intuition as to why [itex] \rho = \sqrt{ \varepsilon \eta } [/itex].
 
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  • #2
##\varepsilon \eta = (r+z)(r-z)=r^2-z^2=h^2## and the height should occur at the position ##(x,y)##.
 

What are parabolic coordinates?

Parabolic coordinates are a coordinate system used to describe points in a two-dimensional plane. They are based on the shape of the parabola and are often used in mathematical and scientific calculations.

How are parabolic coordinates defined?

Parabolic coordinates are defined as (u, v) where u is the distance from a fixed point on the parabola and v is the angle formed by the line connecting the point to the focus of the parabola and the x-axis.

What is the relationship between parabolic coordinates and Cartesian coordinates?

Parabolic coordinates and Cartesian coordinates are related by the following equations:
x = u2 - v2
y = 2uv

How are parabolic coordinates used in physics?

Parabolic coordinates are commonly used in physics to describe the motion of particles in a parabolic potential, such as a particle moving under the influence of gravity. They are also used in electromagnetism to describe the electric and magnetic fields of a charged particle moving in a parabolic trajectory.

What are some applications of parabolic coordinates?

Parabolic coordinates have many applications in mathematics, physics, and engineering. They are used to solve differential equations, calculate electric and magnetic fields, and analyze the motion of particles in a parabolic potential. They are also used in optics, fluid mechanics, and heat transfer problems.

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