# Thread: Uniqueness of Spherical Coordinates

1. ## Uniqueness of Spherical Coordinates

True/false quiz at the end of chapter 14 of Calculus with Analytic Geometry by Purcell and Varberg, #25 reads,

If we restrict $$\rho, \theta, \phi$$ by $$\rho \ge 0, 0 \le \theta < 2\pi, 0 \le \phi \le \pi$$ then each point in three-space has a unique set of spherical coordinates.

I answered false because theta can take any value in [0, 2pi) for any point with phi equal to 0 or pi. Additionally, the origin (rho = 0) allows the same range for theta and also for phi to take any value in [0, pi].

The answer key has true for this statement.

2. ## Re: Uniqueness of Spherical Coordinates

Originally Posted by Zexuo
theta can take any value in [0, 2pi) for any point with phi equal to 0 or pi.
This seems to be a description of the two parameters, not a refutation of anything.

Additionally, the origin (rho = 0) allows the same range for theta and also for phi to take any value in [0, pi].
Now, you're on to something. If $$\rho = 0$$, does it matter what $$\theta\;and\;\phi$$ are? Anything BESIDES the Origin?

3. ## Re: Uniqueness of Spherical Coordinates

As an example, the coordinates (1, 0, 0) and (1, pi/2, 0) refer to the same point P. This would mean that P has more than one set of spherical coordinates.

Or does the statement of the problem mean that each set of spherical coordinates refers to one and only one point?

4. ## Re: Uniqueness of Spherical Coordinates

Originally Posted by Zexuo
As an example, the coordinates (1, 0, 0) and (1, pi/2, 0) refer to the same point P. This would mean that P has more than one set of spherical coordinates.

Or does the statement of the problem mean that each set of spherical coordinates refers to one and only one point?
Typically spherical coordinates are given by $\displaystyle \rho$ as a distance from the origin, $\displaystyle \pi$ an angle measured from the +z axis (so $\displaystyle 0 \leq \theta \leq \pi )$, and $\displaystyle \phi$ an angle measured from the +x axis in the xy plane (so $\displaystyle 0 \leq \phi < 2 \pi )$. If we take these as I defined then the point (1, 0, 0) in rectangular coordinates corresponds to $\displaystyle (1, \pi / 2, 0)$ in spherical coordinates.

And yes, the question is to show that for each rectangular coordinate point there is only one coordinate in spherical coordinates.

-Dan

5. ## Re: Uniqueness of Spherical Coordinates

I've seen the scheme you mention elsewhere. This book uses (rho, theta, phi) with theta the angle from the x-axis to the projection of rho on the xy-plane and phi the angle from the z-axis to the line segment between the origin and the point.

In my example I meant (1, 0, 0) and (1, pi/2, 0) both as spherical coordinates referring to the same point with the Cartesian coordinates (0, 0, 1).

6. ## Re: Uniqueness of Spherical Coordinates

Most mathematics texts will use $\theta$ for the polar coordinate in the $xy$ plane and $\phi$ for the angle from the $z$ axis to the radius vector $\rho$. Unfortunately, apparently it is common in physics to switch the definitions of $\theta$ and $\phi$. I agree with you Zexuo, the question is false as stated. Also consider $(1,\pi, \theta)$ for any $\theta$ in the given range.