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Offline Bobby Shafto

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The Sagitta (Globe Earth "Theory")
« on: January 10, 2019, 06:41:09 PM »
Sagitta is the fancy term for what is commonly referred to in flat vs. globe discussions as "the bulge."



I've always been a bit annoyed by references to "the bulge" in talking about "globe earth theory;" and not because of the term itself but because I don't think the measure of the arc above the chord is germane. The bulge is not a mound or hill that rises up in front of you on a globe and presenting an obstacle. The bulge is not the obstacle. The horizon is.

The only time that the bulge matters is when the horizon happens to coincide with it, as I've depicted with the diagram (from Metabunk's interactive visualizer) above. The horizon is a function of height above the arc. The bulge is a function of base locations on the arc. For any two points on the arc of a globe, the obstructing horizon varies with height but the bulge is fixed.  Only at a height where the horizon corresponds to the midpoint on the arc between two points does the bulge matter, and then only because it's where the horizon is.

The "bulge" isn't a mound rising before your eyes. On a globe, the arcing surface drops away from your eye level. If I had my druthers, we'd banish "the bulge" from any discussion about how things work on a globe. It's often misconstrued and misrepresented as that which causes or would cause an obstruction to line of sight compared to a flat earth; but that's in error.

Consider two points on earth that are 30 miles apart.
On a flat surface, they're just that: 30 miles apart.
On a globe with a radius of 3959 miles, they are 30 miles apart on an arc of 0.43°. The height of that arc above a straight line (chord) between those two points -- that height being the bulge or sagitta -- is 0.0284 miles or 150 feet.  Over the span of 158,400' between the two points on the arc, how significant is that 150' sagitta?

That additional 150' of the sagitta (bulge) contributes some amount of additional distance on the arc of a globe compared to the straight line chord length which would be the flat earth distance. Guess how much extra distance? Or, if you know how to do the math, work it out.

I'll be back.

Edit: Come on, my fellow globelings. This question is for anyone, and I'm sure globe supporters can come up with the solution. There's probably even an online calculator that can provide the number.

But until then, just guess. Anyone? What do you think the difference is between arc length and chord length in the 30-mile distance scenario above that a 150' "bulge" would produce? Just ballpark it.


 
« Last Edit: January 10, 2019, 07:37:33 PM by Bobby Shafto »

Re: The Sagitta (Globe Earth "Theory")
« Reply #1 on: January 10, 2019, 08:35:21 PM »
Using the formula S = r*θ I got 29.7 miles. But you gave us and used rounded figures, didn't you? Using the given radius of the Earth and the given sagitta with Pythagorean theorem to get the half chord, I got a chord value of 28 miles, a tiny bit off of the given 30 miles. So I got an additional distance of 1.7 miles comparing arc length to chord length.

Edit: Using this online arc calculator and the given radius and angle, the chord and arc were almost the same length (probably why you chose it, you also need to increase digits after decimal point to at least 3 or 4 to see the difference), but it appears your sagitta was a little off, which is why my own chord value is probably off. https://planetcalc.com/1421/
« Last Edit: January 10, 2019, 08:41:46 PM by Bastian Baasch »

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Offline Bobby Shafto

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Re: The Sagitta (Globe Earth "Theory")
« Reply #2 on: January 10, 2019, 08:52:03 PM »
Thanks for giving this a whirl. So as not to confuse the issue, don't rely on any of my numbers to work it out. Just use 30 miles on the arc of a circle with radius of 3959 miles.

What's the difference between arc length and chord length? 1.7 miles?

Re: The Sagitta (Globe Earth "Theory")
« Reply #3 on: January 10, 2019, 08:59:59 PM »
Thanks for giving this a whirl. So as not to confuse the issue, don't rely on any of my numbers to work it out. Just use 30 miles on the arc of a circle with radius of 3959 miles.

What's the difference between arc length and chord length? 1.7 miles?

Sorry for needlessly complicating things, but the difference is 0, there is none between the arc length and chord length.

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Offline Bobby Shafto

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Re: The Sagitta (Globe Earth "Theory")
« Reply #4 on: January 10, 2019, 09:09:20 PM »
There has to be some difference. The arc and chord length can't be equal. The arc length must be greater.

If using that calculator and the two values are the same, try changing to smaller units and/or adjusting the precision. If working with miles and the numbers coming out the same, that might suggest the answer is less than half a mile.

Re: The Sagitta (Globe Earth "Theory")
« Reply #5 on: January 10, 2019, 09:12:22 PM »
There has to be some difference. The arc and chord length can't be equal. The arc length must be greater.

If using that calculator and the two values are the same, try changing to smaller units and/or adjusting the precision. If working with miles and the numbers coming out the same, that might suggest the answer is less than half a mile.
Yeah I realized that, the online calculator wasn't good enough, redoing on my graphing calculator, I got a difference of 0.00007178 miles.

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Offline Bobby Shafto

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Re: The Sagitta (Globe Earth "Theory")
« Reply #6 on: January 10, 2019, 09:55:48 PM »
...redoing on my graphing calculator, I got a difference of 0.00007178 miles.
That's 4.5 inches.

Can that be right?  30 miles, and the difference between the distance over a globe earth with a bulge and a flat earth is only 4-5 inches?

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Offline RonJ

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Re: The Sagitta (Globe Earth "Theory")
« Reply #7 on: January 11, 2019, 03:36:54 AM »
OK, Bobbie.  I did the math a couple of different ways and came out with the same 4.54 inches that you guys got.  It did sound a little strange because the hump is 150 feet above the chord.  I have a HP48 calculator and a MathCad program for my computer and both yielded about the same results.  With a triangle of 80,000 feet on one side and 150 feet on the other, the hypotenuse is 80000.1406 feet.  The straight line is about the same as the curved chord in that situation.  I tried that just to see if all the results were at least reasonable, and it looks like they are.   
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Offline Bobby Shafto

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Re: The Sagitta (Globe Earth "Theory")
« Reply #8 on: January 11, 2019, 04:15:24 AM »
Right.

The "bulge" is a useless measure, IMO. Worse, it gets misapplied by those on both sides.

I think I'll make this pitch to Mick West (Metabunk) and Walter Bislin. See what they say since their curvature calculators are so often used.

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Offline Tumeni

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Re: The Sagitta (Globe Earth "Theory")
« Reply #9 on: January 17, 2019, 12:08:34 AM »
Responding to the last paras of the OP, the maths is outlined in the Wiki for 'Circular Segment'

https://en.wikipedia.org/wiki/Circular_segment



For completeness, apply the formulae for a Spherical Cap

https://en.wikipedia.org/wiki/Spherical_cap



 
« Last Edit: January 17, 2019, 12:39:00 AM by Tumeni »
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