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  #51  
Old 02-17-2018, 03:02 PM
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Quote:
Originally Posted by Riemann View Post
I wonder what became of 3_14159265358979323846?
We need one of his friends to come back, bring 3_14159265358979323846.
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  #52  
Old 02-17-2018, 03:09 PM
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Originally Posted by eschereal View Post
"Square" degrees sounds not quite right.
It's not even quite wrong.

But given the original statement, "360 degrees in a circle", it's probably the best you can do. Saying a circle has 360 degrees is only barely a measurement (as opposed to just an arbitrary definition of "degrees"). It tells you nothing about the size of the circle or anything else, it just suggests that from a point at the center of the circle, you could turn 360 degrees and you'd still see the circle. As opposed, I guess, to a circle-shape with a wedge cut out of it (i.e. a Pac-Man shape) where you might say "this has only 330 degrees in it", i.e. a 30-degree wedge is missing.
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  #53  
Old 02-17-2018, 04:42 PM
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It's not correct to say that nobody ever uses square degrees. Astronomers do. Of course, we're usually looking at a small enough patch of sky that the spherical curvature is negligible.
  #54  
Old 02-17-2018, 06:11 PM
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Originally Posted by Chronos View Post
It's not correct to say that nobody ever uses square degrees. Astronomers do. Of course, we're usually looking at a small enough patch of sky that the spherical curvature is negligible.
And I bet small enough patch that SI prefixes to radians would be less elegant and non-rational with compounding losses of precision.

In computers this a problem 0.1 = 1/10, and 10 is not a power of two. There will be an representation error without tricks which produces an irrational number and it may not symmetric for equivalence tests.

1 min arc = 1/21,600 of 1 turn = pi/10800 radians
1 arcsecond = 1/60 arcminute = 1/3600 degree = 1/1,296,000 turn = pi/648000 radians.


Probably not as big of an issue if you are calculating watts per steradian for RF or lasers but I can see why Astronomers would like the option of staying with rational numbers.

Last edited by rat avatar; 02-17-2018 at 06:13 PM.
  #55  
Old 02-17-2018, 08:31 PM
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I think we all agree with 3_141592653589...'s answer, but the number of "degree confluences" may also have interest:
Quote:
Originally Posted by CurtC View Post
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.
Of these 64,442 confluence points, 21,543 are on land, 38,409 on water, and 4,490 on the Antarctic and Arctic ice. There are 52 confluence points in California; the three in the Bay Area are near Santa Cruz, Concord, and the Point Reyes lighthouse. There's one near Cinnaminson, Philadelphia. The confluence point nearest to me is a few miles south of an American-owned spaghetti sauce factory. Visit the Degree Confluence Project to help document the confluence point near you.

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Originally Posted by ultrafilter View Post
Bear in mind also that whatever method you come up with for spheres has to work for more general solids.
What does this mean? What's the 2-D analogy?
  #56  
Old 02-17-2018, 10:11 PM
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rat avatar, rationality isn't really an issue. Sure, something that's a rational number of square degrees will be an irrational number of steradians, but the reverse is also true: Something that's a rational number of steradians will be an irrational number of square degrees. And most contexts where it'd come up would be equally likely to be irrational in either units: That is to say, almost guaranteed, and so you just round it to whatever number of digits is convenient anyway.
  #57  
Old 02-17-2018, 10:16 PM
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I think you have to define "cubic degrees" in terms of hyperspheres.
Seriously, is there a hypersphere equivalent to radians, that defines the volume of a sphere in terms of the surface of a hypersphere?
  #58  
Old 02-17-2018, 11:34 PM
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Looking at the blue line on this graph you see 2π radians at 2D, 4π steradians at 3D, 2π2 at 4D, all the way up to π6/60 at 12D.
  #59  
Old 02-18-2018, 12:36 AM
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Quote:
Originally Posted by Chronos View Post
rat avatar, rationality isn't really an issue. Sure, something that's a rational number of square degrees will be an irrational number of steradians, but the reverse is also true: Something that's a rational number of steradians will be an irrational number of square degrees. And most contexts where it'd come up would be equally likely to be irrational in either units: That is to say, almost guaranteed, and so you just round it to whatever number of digits is convenient anyway.
I should have clarified that sticking with SI and prefixes is the issue.

The milliradian, mil, or mrad works fine in optics, but that style of prefixing would lead to issues going with a pure SI prefix model...

Consider parallax:

d = 1/p

And then the implications for the parsec, which would require trig to avoid FP loss of precision on small numbers or fighting with representation errors in binary.

Sure the above has some errors due to using sine small-angle approximation, but that error is 0.01ppm below one degree and gets smaller. That said I am focused on the limitations of computers.

A couple of other quick FP issues that make it nice to have sub units that are still rational.
Subtracting similar numbers leads to a loss of precision, Which due to the cumultive effect caused this Patriot missile failure.

tan(π/2) will not return infinity or an error.

Addition and Multiplication will not be reliably associative so: (foo + bar) + baz != foo + ( bar + baz ). This the big one I would worry about if we were to move everything to SI prefixes for magnitude to get rid of Deg/Min/Sec.
Obviously we use floats for lots of numbers, but there is a lot of value in avoiding irrational numbers when possible.
from decimal import Decimal

Decimal.from_float(0.1)
Out[25]: Decimal('0.1000000000000000055511151231257827021181583404541015625')

Decimal.from_float(0.01)
Out[26]: Decimal('0.01000000000000000020816681711721685132943093776702880859375')

Decimal.from_float(0.001)
Out[27]: Decimal('0.001000000000000000020816681711721685132943093776702880859375')

Decimal.from_float(0.0001)
Out[28]: Decimal('0.000100000000000000004792173602385929598312941379845142364501953125')

Decimal.from_float(0.0000001)
Out[29]: Decimal('9.99999999999999954748111825886258685613938723690807819366455078125E-8')
The point is that degrees, min, sec will not suffer this issue as much if you can stay rational, With arc seconds you can pi until you need it too use it.

Last edited by rat avatar; 02-18-2018 at 12:39 AM.
  #60  
Old 02-18-2018, 06:55 AM
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The loss of precision you're describing will never be relevant for actual astronomy, and even if it were, the solution would be to use higher precision floats.
  #61  
Old 02-18-2018, 07:48 AM
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Issues with loss of precision are caused by bad or misapplied numerical algorithms; the choice of units (degrees versus radians) or whether the quantity being approximated is rational or irrational should not make a difference.

As for hyperspheres, there is no reason to stop at any number of dimensions: area and volume are defined in Euclidean n-space and the surface area of a unit sphere is exactly 2πn/2/Γ(n/2). In fact, inspecting this formula shows that as the number of dimensions grows the surface area tends to zero, even though in low dimensions it looks like it's growing. Also, despite this, the ratio of volume to surface also gets smaller (e.g., the area of a disc is 1/2 of its radius times its circumference, but for a sphere the ratio is only 1/3, and so on.) I suppose that actually using hyperradians to measure things is rare in practice, though I would love to hear about such cases.
  #62  
Old 02-18-2018, 07:48 AM
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Quote:
Originally Posted by rat avatar View Post
And I bet small enough patch that SI prefixes to radians would be less elegant and non-rational with compounding losses of precision.

...Probably not as big of an issue if you are calculating watts per steradian for RF or lasers but I can see why Astronomers would like the option of staying with rational numbers.
I'm an astronomer and I have no idea what you're talking about. When astronomers use degrees, arcminutes, arcseconds etc, etc, it's because of tradition, the same reason the US still uses inches. We keep using it because that's what we learned in college. As far as computational accuracy is concerned , there is no benefit to a full circle being a rational number of degrees. Units are used for measuring many different things, most of which aren't a rational number of degrees or steradians. And even if one of your numbers is rational, you lose that advantage as soon as you actually use that number in a calculation, because the next intermediate result will not be rational.

And my colleagues and I generally use steradians for radiometric calculations. Though I've used square arcseconds in a few situations because that gives a result that's easier to interpret. (For observing a small astronomical object, "10 photons per second per square arcsecond" is easier to interpret than "2e11 photons per second per steradian".)

Last edited by scr4; 02-18-2018 at 07:50 AM.
  #63  
Old 02-18-2018, 08:14 AM
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Take a circle and spin it on it's axis half a turn to get a sphere.

My answer is 360 deg mark 180 deg.
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  #64  
Old 02-18-2018, 12:47 PM
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Originally Posted by Enola Straight View Post
Take a circle and spin it on it's axis half a turn to get a sphere.

My answer is 360 deg mark 180 deg.
In other words, 360x180=64800? That's the number of grids on the surface of the Earth where the sides are latitude and longitude lines exactly one degree apart. But these grids vary in size, getting smaller as you leave the equator. At the equator, they resemble squares. But at the poles they are extremely skinny triangles that don't even resemble squares at all.

Try taking one such square at the equator, one degree of latitude by one degree of longitude. It's a square 111.3 km on a side. That's about 12,388 km2. The total area of Earth is 510.1 million km2, so that's a ratio of about 41,160. If you could come up with a way to tile the surface of the Earth with 41,160 sectors of equal size, each one of them would be the same size as a one-degree-by-one-degree square at the equator. One option would be to adjust their north-south heights as you move away from the equator, while keeping the widths at exactly one degree of longitude. That gives sectors that resemble trapezoids, 360 of them in each row, and we'd need about 114 rows (57 up from the equator to the north pole and 57 down from the equator to the south pole) for a total of 41,040 sectors. This would cause some headaches for navigation because the tops and bottoms aren't one degree of latitude apart. But each sector would be the same size, approximately 1 square degree.

That's if you want to try to stick with "degrees". The logical alternative is to switch to radians.

Imagine a piece of rope the same length as your arm. Now swing your arm in a complete circle. How many ropes would it take the cover the circle traced by your finger tip? Just over six. That's why there's 6.283 radians in a circle.

Imagine a large birthday square birthday cake where each side is the same length as your arm. Now swing both arms to trace a sphere. How many birthday cakes would it take to cover the sphere? Just over 12 1/2. That's why there's 12.566 steradians in a sphere.

I say the answer is 12.566 steradians in a sphere (or 2 times tau if you want to be precise).

Last edited by sbunny8; 02-18-2018 at 12:49 PM.
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