Polar Coordinates: A Nicer Way to Define?

In summary, the polar coordinates of a point is defined by the rectangular coordinates of that point according to the transformation T from R² to R² defined by the bijection x=r\cos\theta, y=r\sin\theta. However, this definition fails for y=pi and x=2 because tan(pi/2) is not defined. The solution is to define the polar coordinates through T:(x,y)\mapsto (\sqrt{x^2+y^2},arctan\left(\frac{y}{x}\right)) for (x,y) \in \{(x,y) \ \vert \ y/x = (n+1
  • #1
quasar987
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As I understand it, the polar coordinates of a point is defined by the rectangular coordinates of that point according to the transformation T from R² to R² defined by

[tex]T:(x,y)\mapsto (\sqrt{x^2+y^2},tan\left(\frac{y}{x}\right))[/tex]

But this definition fails for y=pi and x=2 because tan(pi/2) is not defined.

We could defined it this way for [itex](x,y)\in \mathbb{R}^2 \backslash \{(x,y) \ \vert \ y/x = (n+1/2)\pi, \ n\in \mathbb{Z}\}[/itex], and by

[tex]T:(x,y)\mapsto (\sqrt{x^2+y^2}, Arccos_n \left(\frac{x}{\sqrt{x^2+y^2}}\right))[/tex]

for [itex](x,y) \in \{(x,y) \ \vert \ y/x = (n+1/2)\pi, \ n\in \mathbb{Z}\}[/itex] and where Arccos_n is the inverse function of cos in the interval containing (n+1/2)pi.. i.e. [itex]Arccos_n(z): [-1,1]\rightarrow [n\pi, (n+1)\pi][/itex].

This is phenomenally ugly. Is there a nicer way to define the polar coordinates?
 
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  • #2
quasar987 said:
[tex]T:(x,y)\mapsto (\sqrt{x^2+y^2},tan\left(\frac{y}{x}\right))[/tex]
That has never been correct, this is somewhat more correct:
[tex]T:(x,y)\mapsto (\sqrt{x^2+y^2},arctan\left(\frac{y}{x}\right))[/tex]
 
  • #3
Oh damn!

But more like 'phew!' really. :smile:
 
  • #4
These basis changes should be made through diffeomorphisms.Obviously

[tex] \hat{T}:(x,y)\mapsto \left(\sqrt{x^{2}+y^{2}},\arctan \frac{y}{x}\right) [/tex]

is not a diffeomorphism from [tex] \mathbb{R}^2}\rightarrow \mathbb{R}^{2} [/tex].

Daniel.
 
  • #5
What does that mean Daniel?
 
  • #6
Invertible [itex] C^{\infty} [/itex] maps from one space to another.

U'll have to exclude the Oy axis.

Daniel.
 
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  • #7
Well I don't see your point. Why must the transformation from rectangular to polar coord. be a diffeomorphism absolutely?!
 
  • #8
Pick the point [itex] (0,2) [/itex] in cartesian.Can your mapping send it to polar coords...?

Daniel.
 
  • #9
No. So how do you suggest we avoid this problem?
 
  • #10
It's simplest to define the mapping by through:
[tex]x=r\cos\theta, y=r\sin\theta[/tex]

This sets up a bijection almost everywhere between (x,y) and [tex](r,\theta)[/tex].
(That is with (x,y) on the plane, r on the non-negative half-axis, and [tex]\theta[/tex] on the half-open interval [tex][0,2\pi)[/tex]
 
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  • #11
If theta is traped in the interval [0, 2pi), then surely when we say that the potential energy of an object free to rotate around the z axis is

[tex]V(\theta) = -\int_{\theta _s}^{\theta}N\z(\theta)d\theta[/tex] (Symon pp.212)

the theta involved in this equation is not the theta of polar/cylindrical coordinates (i.e. constrained in [0, 2pi)), is it?

I had succeeded in proving this equation but it involved treating the polar angle has being free to take any value in [itex](-\infty, \infty)[/itex]. I was trying to justify that it was justified to do that. But now I'm a little confused. Is it justified?
 

Related to Polar Coordinates: A Nicer Way to Define?

1. What are polar coordinates?

Polar coordinates are a way of defining points in a two-dimensional plane using a distance from the origin and an angle from a reference line.

2. How are polar coordinates different from Cartesian coordinates?

Polar coordinates use a distance and angle to define a point, while Cartesian coordinates use two perpendicular axes (usually x and y) to define a point.

3. What are the advantages of using polar coordinates?

One advantage of polar coordinates is that they can more easily represent points in circular or symmetrical patterns, such as in polar graphs or in describing the motion of objects in a circular path.

4. How do you convert between polar and Cartesian coordinates?

To convert from polar to Cartesian coordinates, you can use the formulas x = rcosθ and y = rsinθ, where r is the distance from the origin and θ is the angle from the reference line. To convert from Cartesian to polar coordinates, you can use the formulas r = √(x^2 + y^2) and θ = tan^-1 (y/x).

5. How are polar coordinates used in real-world applications?

Polar coordinates are commonly used in physics and engineering to describe circular motion and in navigation systems to determine the position of objects relative to a central point. They are also used in astronomy to describe the locations of stars and other celestial objects.

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