Eli5:In non-euclidean geometry (for my example a globe).Lines of longitude are parallel @ the equator but meet @ the poles. Lines of latitude (let’s take tropic cancer, equator & tropic Capricorn)also 3 lines are parallel, also on a curved plain, but this time don’t converge.What am I missing out?

137 viewsMathematicsOther

Eli5:In non-euclidean geometry (for my example a globe).Lines of longitude are parallel @ the equator but meet @ the poles. Lines of latitude (let’s take tropic cancer, equator & tropic Capricorn)also 3 lines are parallel, also on a curved plain, but this time don’t converge.What am I missing out?

In: Mathematics

4 Answers

Anonymous 0 Comments

Lines of longitude are all exactly equal in length and are “great circles” – the longest possible line you can draw on a sphere. Every great circle will intersect every other great circle in two places. The only line of latitude that is a great circle is the equator. All other longitudinal lines are shorter. You can have an infinite number of non-intersecting circles on a sqare, provided you have at most one great circle.

Anonymous 0 Comments

The lines of longitude are all circles of the same size, rotated slightly so they only cross at the poles. They are only parallel at the equator because at all other points they are getting closer as they converge.

The lines of latitude are concentric circles; different sizes next to one another. That means they don’t overlap.

Anonymous 0 Comments

Because “lines” refers to straight lines. There are no 3d straight lines on a globe, but there are straight lines within the 2d surface of the globe. If you drove a car along a line of latitude you’d have to turn the steering wheel a little bit (unless it’s exactly the equator) — this is easy to see if you consider a line of latitude near one of the poles.

In short, lines of latitude aren’t straight lines within the geometry of a sphere. Straight lines on a sphere are those circles that split the sphere into two parts of equal size — also know as “great circles”. Two different great circles do indeed always intersect.

Anonymous 0 Comments

A “line” in spherical geometry are the [great circles](https://en.wikipedia.org/wiki/Great_circle) – what you get when you intersect a plane through the center of the sphere. They give us things like the shortest distance between two points, which is what regular lines do in Euclidean geometry.

The only ~~longitude~~ latitude line that’s *actually* a “line” in the sense of spherical geometry is the equator. All the other latitudes should be thought of as curves rather than lines. This is why they don’t intersect, which is what any two pairs of lines in spherical geometry do. You should see their non-intersection as being just as relevant as [these two parabolas not intersecting](https://www.desmos.com/calculator/x57ji80vhq).