JohnnyLA, I think you’re a pilot, so think of it in terms of navigation. You know that one degree of longitude at the equator is around 60 nautical miles. Also, one degree of latitude is also about 60 nautical miles. So one square degree on the earth’s surface corresponds to about 3600 square nautical miles. So how many square degrees are there on the earth? You can take the number of square nautical miles, and divide by 3600. The radius of the earth is around 3440 nmi, so the surface area is 4*pi times that: 148,700,000 nmi^2. Dividing by 3600 gives 41,307 square degrees.
As you take smaller and smaller angles, the answer will get closer and closer to the 41,253 number that 22/7 gave above.
I get the feeling we’re talking about two different things. Yes, it’s true that at the equator one degree of longitude is 60nm and one degree of latitude covers less distance the farther away from the equator you go. But regardless of the distance it’s still a degree.
Suppose you have a globe and designate the north pole as the origin. Arbitrarily set one “great circle” as the prime meridian. Okay, now you can go one degree south and one degree west. Your coordinates from the pole would be 1, -1, right? Okay, you go all the way west and you’ll have 360° around the first degree south of the pole. Now drop down another degree from the pole so that your longitudinal coordinate is -2. Go around the globe at -2 and you get 360 more degrees. And so on. The area covered by the triangle formed between the first degree down, one degree over, and the pole will be different from the area formed by the quadrangle formed by the area between -1 and -2 and 1° west. But it doesn’t matter what the difference is because we’re not talking about area; just the number of whole degrees.
So I’m not talking about “square degrees”, but the number of points seperated by one degree on a sphere. Starting from the pole and going to the equator, you have 90 “layers” that are one degree apart. Each of those “layers” have 360 points along its edge that is one degree apart – regardless of the linear distance between them. From the equator to the south pole there are another 90, 360° layers.
So do you see what I’m saying? Are we talking about two different things?
When you say that “there are 360 degrees in a circle,” you aren’t really stating a fact about circles, but about degrees: a degree is, by definition, 1/360th of the way around a circle.
You can specify exactly where you are on a given circle with only one number (the angle, in degrees or some other unit), but it takes two coordinates to tell where you are on a sphere, so a sphere is a two-dimensional surface (though it lives in three-dimensional space).
One way of specifying points on a sphere is by using spherical coordinates (as referred to by Rabid Squirrel), which involve two angles: theta, between 0 and 360 degrees, which is like longitude on the Earth’s surface, and phi, which ranges from 0 at the north pole, through 90 degrees at the equator, to 180 degrees at the south pole. (A third spherical coordinate, r, represents a point’s distance from the origin, but this would be constant for all points on a particular sphere.)
We’re using a different definition of “separated by one degree.” You’re dividing the sphere into a 360x360 grid and if you do that, of course you end up with 360x360 grid elements. By that definition, “one degree” is always one grid element.
But if you use that definition, if you have an island on the equator 60x60 nautical miles in size it’s 1 square degree, but if the same island were at 60 degree latitude it’ll be 2 square degrees. So this “square degrees” isn’t a useful unit for measuring the “angular area” of an object (i.e. how big a fraction of the entire sphere it occupies). Solid angle is a measure of angular size (angular area, sort of) and must be independent of direction. Solid angle is used to measure things like the field of view of a camera or beam size of a lightbulb - you can’t have the number change just because you pointed the lightbulb at the celestrial north pole.
Right. But the question was “how many degrees are there in a sphere?” I took the question literally and visualized global coordinates without regard to the area covered by a set of degrees, and instead just counted points. (Not being a mathematician and not having had any math classes since high school, that’s the only way I could do it. )
So. If you look at a globe and count the points one degree apart as I did the previous post, is the answer of 360° in a circle x 180º of latitude = 64,800 one-degree points on a sphere (globe) correct?
Almost correct. If you painted latitude and longitude lines on the earth, one degree apart, they would intersect at 64,442 locations. At latitudes 1 through 179, there would be 360 points each, plus one point each at latitudes 0 and 180: 179*360+2=64,442.
However, I’m not sure why this is useful information.
Again, we seem to be confusing two definitions of “one degree apart.” Consider two points: [0.0 E, 60.0 N] and [2.0 E, 60.0 N]. If you were looking at a grid pattern on a map, you would call that 2 degrees apart. But imagine drawing a line from each point to the center of the globe and measuring the angle between the two lines; this angle will be 1 degree. Using the latter definition, there are 41,253 points on a sphere which are “one egree apart” from each other.
Hi. Thanks for all your responses. It will take me a bit to assimilate the information you have provided. I’ll re-read and then re-read again and come back with my doubts, if any.
I think the answers from 3_14159265358979323846 and Rabid_Squirrel was what I was looking for. But I’m not quite sure anymore Also, thanks to scr4 for taking the time to answer Johnny L.A.'s questions.
If you have further inputs please do add to the discussion. I’ve learnt quite a bit from your answers so far.
I mean it as you correctly understood it… “What is the number of planes that exist that contain the center point of a given sphere”
Why is this an infinite figure ? Is it something similar to “How many points on a given line segment” ?
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Also, thanks to scr4 for taking the time to answer Johnny L.A.'s questions.