Spherical Coordinates Question

[eyesontheball1]

Homework Statement

I'm feeling a bit ambivalent about my interpretation of spherical coordinates and I'd appreciate it if someone could clarify things for me! In particular, I'd like to know whether or not my derivation of the coordinates is legitimate.

Homework Equations

Considering only the xy-plane, x = rcos(θ), y = rsin(θ) s.t. r ≥0, -π≤θ≤π.

Now, if we introduce the z-axis, so that we're in 3-dimensional space, we can construct a right triangle s.t. the base of the triangle lies in the xy-plane and the right angle of the triangle is formed between the base of the triangle and the side of the triangle pointing upward into the positive z direction normal to the xy-plane. The hypotenuse of the triangle lying in the xy-plane has length r. This hypotenuse is also the base of the second triangle. We choose ø s.t. -π/2 ≤ ø ≤ π/2, and we let r = pcos(ø), so that the length of the base of the second triangle is r = pcos(ø), and we also let the length of the side of the triangle normal to the xy-plane be z = psin(ø). It then follows that the length of the hypotenuse of the second triangle is p. What I'm unsure about is whether or not it's okay to interpret the angle, ø, as being the angle formed between the xy-plane and the hypotenuse of the second triangle, given a fixed point, (x,y), in the xy-plane. Thanks in advance!

The Attempt at a Solution

 

[slider142]

That's fine. Normally, people set the domain of your ø variable to be [0, π], due to the range of the inverse cosine function, but yours covers the same space.

 

[Ray Vickson]

Homework Statement

I'm feeling a bit ambivalent about my interpretation of spherical coordinates and I'd appreciate it if someone could clarify things for me! In particular, I'd like to know whether or not my derivation of the coordinates is legitimate.

Homework Equations

Considering only the xy-plane, x = rcos(θ), y = rsin(θ) s.t. r ≥0, -π≤θ≤π.

Now, if we introduce the z-axis, so that we're in 3-dimensional space, we can construct a right triangle s.t. the base of the triangle lies in the xy-plane and the right angle of the triangle is formed between the base of the triangle and the side of the triangle pointing upward into the positive z direction normal to the xy-plane. The hypotenuse of the triangle lying in the xy-plane has length r. This hypotenuse is also the base of the second triangle. We choose ø s.t. -π/2 ≤ ø ≤ π/2, and we let r = pcos(ø), so that the length of the base of the second triangle is r = pcos(ø), and we also let the length of the side of the triangle normal to the xy-plane be z = psin(ø). It then follows that the length of the hypotenuse of the second triangle is p. What I'm unsure about is whether or not it's okay to interpret the angle, ø, as being the angle formed between the xy-plane and the hypotenuse of the second triangle, given a fixed point, (x,y), in the xy-plane. Thanks in advance!

The Attempt at a Solution

The USUAL convention for spherical coordinates is: the angle between $\vec{r}$ and the z-axis is $\theta \in [0, \pi]$, with $\theta = 0$ being the +z axis. The counterclockwise angle from the +x axis is $\phi \in [0,2 \pi]$; counterclockwise means that $\phi = 0$ along the +x axis, $\phi = \pi/2$ along the +y axis, etc. Of course, these are just conventions and you are not required by law to follow them, but they are the most common definitions. See, eg., http://en.wikipedia.org/wiki/Spherical_coordinate_system . (However, the http://mathworld.wolfram.com/SphericalCoordinates.html site employs your convention.)

Anyway, in this standard convention the transformation equations are
$$x = r \sin(\theta) \cos(\phi)\\<br /> y = r \sin(\theta) \sin(\phi)\\<br /> z = r \cos(\theta) \\<br /> 0 \leq \theta \leq \pi, \;\; 0 \leq \phi \leq 2 \pi$$
Basically, if the +z axis goes through the North Pole and the +x axis passes through Greenwich, England, the angle $\theta$ measures latitude down from the North Pole and $\phi$ measures longitude going East from Greenwich.

 

[eyesontheball1]

Thanks, guys! I just prefer my above derivation of the coordinates because the reasoning used to choose the parameters for the second triangle is identical to the reasoning used to choose the polar coordinates corresponding to the first triangle.