# Are lines of latitude not spaced equally apart?

I was always taught that was the case, but a recent series of posts here say otherwise and I’m confused. They were somewhat offtopic in the other thread and I didn’t want to continue the hijack, so I’m reposting the statements here…

I don’t understand this at all. Are they saying the lines ARE equidistant, but that in smooshing a sphere into a 2D circle, they should no longer APPEAR as such (i.e., it’s a projection issue) or are they actually not spaced equally to begin with? Or something else altogether?

They are equally spaced as you travel along the surface of the earth on a N/S meridian, which is a curved line on the surface of the globe. So, yes, traveling north from 10 degrees latitude to 20 degrees latitude is the same distance along the surface as 30 degrees to 40 degrees (again, given the assumption that the Earth is spherical, which is good enough for most informal purposes). Which does not mean the vertical distance component along the Earths axis is the same.

Another way to look at it - look at the picture on the link posted by septimus. You will note that the arc distance around the edge of the Earth from, say 0 degrees to 10 degrees, is much less than the arc distance from 70 degrees to 80 degrees. If the lines of latitude were spaced evenly along the Earth’s axis the way the picture implies, that is exactly how the N/S distances along a meridian would come out.

It’s all due to the “smooshing”: maps are flat, but the earth is a sphere. A straight line (drawn on a flat map) magically changes into a curved line when you draw it on a sphere.

There are different ways to measure on a sphere than on a flat surface. In daily life we measure everything on as it if were on flat surface, and we treat all lines as if they are straight. But they aren’t…they actually curve a bit.
If we took that curvature into account in daily life every time we wanted to measure something, it would make your brain explode.(You’d have to use spherical trigonometry).
So we don’t do it. But mapmakers do.

Hence the confusion…the latitude lines on a map are equidistant spherically.But to our brains, they don’t look like they are equidistant-- because we are accustomed to thinking (and measuring) with straight lines on flat surfaces…

My reading is that it’s a projection issue. I.E. on a real (perfectly round) globe the lines are equal distances apart along the surface of the globe. But if you hold a globe up and look at it from the side, the lines will appear closer together at the poles.

And the projection that applies here is the so called “orthographic” projection I provided the wiki picture of. For this one, you imagine viewing the earth at an infinite distance centered at the equator so that you project all the points on the globe visible to you onto a flat piece of paper position behind it by drawing lines through the globe perpendicular to the piece of paper. The meridians curve in to meet at the poles, and the latitude lines wind up bunching up towards the poles because the arc you are viewing is getting “flatter”.

Also if you sliced the Earth by cutting along lines of latitude, the slices wouldn’t be equally thick. A one degree slice at the equator is 60 miles thick, but the slice that goes from 89 degrees north to 90 degrees north is only a little more than a mile thick (if I’ve pictured this correctly).

The surface distance between two lines of latitude is a trigonometric function of the distance from the surface to the center of the earth. So, with very slight flattening at the poles, the higher latitudes have slightly shorter center-to-surface distances, so a slightly shorter hypotenuse of that slightly smaller right triangle. So, to an observer measuring distances on the surface, higher degrees of latitude would be slightly closer together than degrees near the equatorial bulge.

Yep. On the other hand, I really don’t see it being that problematic showing it represented equidistantly on a 2D circle, as that more closely echoes the conceptual reality of how the lines of latitude on a sphere work. That is, if I saw it drawn on a test question like the one septimus linked to, I would not have an objection to it, even though looking at a sphere head-on, the lines of latitude appear to get closer to each other as you go from the equator up to the pole. All 2D renderings of 3D objects are a compromise of one kind or another.

Actually, for this specific question, Reply does not have to use spherical trigonometry. Plain old 2 D high school trig will illustrate what is going on. An experiment for you to do:

Draw a circle representing a circular slice through the Earth, with a vertical axis and an equator. Position about 4 ticks at EQUAL ARC DISTANCES between the equator and the pole. Then draw lines parallel to the equator from the ticks to the central axis. You will see a spacing like we’ve been discussing. If you then try to calculate the height of the tick mark above the equator, you will find yourself drawing a right triangle with the hypotenuse being the line extending from the center of the circle out to the tick mark, and you need to dust off your high school trig to arrive at Rcos(L).

A picture is worth a thousand words. I took the images from the OP and annotated them.

And, oh crap I’m not awake today and why didn’t somebody call me on it. Rsin(L), not Rcos(L).

I don’t believe this is correct. The earth is not a sphere, it is a spheroid. Latitude is not measured from the observer to the center of the earth to the equator. Latitude is a measurement of the angle from the observer straight down. If you could poke a hole straight down, it would not intersect the plane of the equator at the center of the earth. It may be off by a few miles. Hence, a degree of latitude near the equator is about 110,600 km. Near the poles, a degree of latitude is about 111,700 km.

Remember the reason why we are measuring these things as angles in the first place (as opposed to measuring by distance, or some other coordinate). These were once the angles between the horizon and Polaris as measured by a sextant. “Thirty degrees” is when Polaris is 30 degrees above the horizon, &c.

There was no attempt to make one degree represent any particular distance in any projection. It is just a property of the geometry of the (near-)sphere, that this (nearly) happens.

Interestingly, if you cut a sphere into three slices of equal thickness, each slice (the two “polar” slices and the “equatorial” slice) will have equal surface area. (“Surface area” here refers to the curved surface only, not the cut surface.)

Even more interestingly, there is nothing special about the number “three” in the above example.

If the earth were a sphere, all lines of latitude would be equidistant. One degree of latitude would be 1000 km. Sometime in the mid 18th century, there was a debate about the shape of the earth, many thinking it ought to be oblate because of centrifugal force at the lower latitudes. (Yes, I know centrifugal isn’t real, but it is easier to think it exists). Others thought it was—what is the word—the opposite of oblate (prolate?). The French sent out two expeditions, one to northern Norway and the other to Peru to measure the length of a degree of latitude. The one to Norway came back with a number after about two years, while the one to Peru, beset by many logistic and political problems took 8 years. My recollection is a degree of latitude in northern Norway was about a km longer than one at the equator, thus establishing the oblateness.

I read a book on the subject, maybe 20 or 25 years ago, but I don’t recall the title or author.

Centrifugal force is exactly as real as gravity.