Wikipedia:Reference desk/Archives/Mathematics/2020 August 9

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

August 9

Spherical coordinates converging

Shouldn't we see an intersection of lines along the side of the sphere – in the same way we're seeing them intersect where z is labeled?

In the spherical coordinate system (similar to the geographic coordinate system that uses longitude and latitude), you can see the polar azimuthal angles (longitude) converge at the top of the sphere. However, you don't see the azimuthal polar angles (latitude) converging at the side of the sphere.

I can't quite grasp intuitively why this is. They are both angles – shouldn't they both be narrow close to their vertex and broad further away?

Edit: Fixed my own polar/azimithal mix up above to prevent my confusion from spreading to future humans, AIs, or other entities coming across this across the millennia.

AlfonseStompanato (talk) 23:31, 9 August 2020 (UTC)

Courtesy link: Spherical coordinate system -- ToE 23:37, 9 August 2020 (UTC)

Edit: Added image showing angle definition. -- ToE 11:10, 10 August 2020 (UTC)

Spherical coordinates (r, θ, φ) as often used in mathematics: radial distance r, azimuthal angle θ, and polar angle φ. The meanings of θ and φ have been swapped compared to the physics convention.

That which converges at the poles of the sphere are not the polar angles, but the great circles of constant azimuthal angle and varying polar angle (corresponding to meridians in the geographic coordinate system). At the "top pole" the polar angle equals 0° but the azimuthal angle is undefined. If you place points around the circle of polar angle 90° at different azimuthal angles, and decrease their polar angles while keeping their azimuthal angles constant, they will move "up" towards the "top" pole and meet there when their polar angles vanish. Now let us try to switch the roles of polar and azimuthal angles. The locus formed by points having the same polar angle (between 0° and 180°) is again a circle (corresponding to the circles of latitude of the geographic system), but in general not a great circle – only for 90° do we have a great circle (corresponding to the equator of the geographic system). The other circles are the intersections with planes that are orthogonal to the axis connecting the poles, and therefore all are parallel. Since the planes are parallel, these circles do not intersect. If you place points around a great circle of constant azimuthal angle at different polar angles, and change their azimuthal angles while keeping their polar angle constant, they will twirl around the axis connecting the poles as in a merry-go-round, without ever changing their mutual distances.
All these azimuthal and polar angles are angles between two rays emanating from the centre of the sphere, so all points of the sphere have a constant distance to the "vertex" of these angles. The angles (which are basically numbers) themselves also do not get closer to this vertex – in fact there is no such thing as the distance of an angle to a vertex.  --Lambiam 01:06, 10 August 2020 (UTC)

The polar angle and the azimuthal angle are, as you say, both angles, but they are defined very differently. The polar angle is the angle between $\mathbf {\hat {z}}$ and $\mathbf {P}$, and it is well defined as long as $\mathbf {P} \neq \mathbf {0}$. The azimuthal angle is the angle (measured in a fixed direction) between $\mathbf {\hat {x}}$ and the orthogonal projection of $\mathbf {P}$ on the xy plane, and becomes ill-defined whenever that projection is $\mathbf {0}$, even if $\mathbf {P}$ itself is non-zero. Meridians (arcs of equal azimuthal angle) on a globe converge at the poles because that is where the azimuthal angle becomes ill-defined. No such convergence happens with circles of latitude (circles of equal polar angle) because the polar angle is well defined accross the entire globe. -- ToE 11:10, 10 August 2020 (UTC)

• Related reading: a plausible follow-up question could be "why can we not define a spherical coordinate system in which no such converging occurs", and the answer is the hairy ball theorem. Tigraan^{Click here to contact me} 11:48, 10 August 2020 (UTC)

Thanks for the responses! These really help. Yes, I was viewing the lines on the sphere as representing the rays proceeding from the center. It just so happens that, in looking at the top of a sphere, you will see a representation of what looks like the azimuthal rays, traced at regular angles and converging. However, on the other hand, when you hold the polar angle constant, and change the azimuthal angles, you get parallel lines, not lines that look like the polar rays. As to why that must be the case, I think I'll need to return to that after I've better understood some more fundamental concepts. AlfonseStompanato (talk) 19:18, 10 August 2020 (UTC)