That's how I started out: plunging into 3-D geometry. Then I realised I could replace the horizon with the great circle passing through the straight-edge, giving a simpler treatment valid for an observer at the earth's surface. I was pleasantly surprised that this lost the need not only for my height above sea level but the distance of my eye from the straight-edge.
Thank you for your derivation / justification of an approximation based on summing finite terms of an infinite series, which can at least serve to reassure us that (_1&o.) is not exhibiting grossly anomalous behaviour near the limit of its range. That approach is applicable to the trig formula, too… I would like to go back to a trigonometric treatment, because that seems to offer better scope for varying the height of the observer above the earth's surface. Which is better suited to the task of offering advice to "Mad" Mike Hughes, the flat-earth rocketeer mentioned in the original article, on how far he needs to ascend to perceive the earth's curvature with his own eyes. And replacing (_1&o.) and (2&o.) with series expansions would permit a better J phrase for D preserving extended precision throughout. On Tue, Feb 27, 2018 at 5:24 AM, J. Patrick Harrington <[email protected]> wrote: > Indeed. The (ideal) distance to the horizon (the point where your line of > sight is tangent to the spherical ocean surface) depends upon your height > above the surface of the ocean. That would have to enter into your formula. > The curvature would be easy to see from the space station... > > Patrick > > > On Mon, 26 Feb 2018, Raul Miller wrote: > >> You might also want to take into consideration the height of your eyes >> above the surrounding terrain. >> >> This might not seem like much, if you are standing on flat ground (for >> example, pretty much anywhere in Maryland (USA - not the long since >> vanished country in Africa)). However, if you're standing on a >> mountain it can be a significant distance. >> >> -- >> Raul >> >> >> On Mon, Feb 26, 2018 at 9:52 PM, Ian Clark <[email protected]> wrote: >> >>> In "The Curious Incident of the Dog in the Night-Time", Mark Haddon has >>> his >>> young autistic hero verify that the earth is not flat by holding up a >>> steel >>> straight-edge to the horizon. >>> >>> In the course of private research into public gullibility, I tried to >>> replicate Haddon's thought-experiment – and failed! Try as I might, I >>> could >>> not see the slightest discrepancy between the horizon and my >>> straight-edge. >>> >>> I strongly suspect that Haddon hasn't performed the experiment himself, >>> but >>> simply written down what he thought one "ought" to see. >>> >>> So what am I to conclude? >>> (a) The earth is flat. >>> (b) The curvature is too small to be seen this way. >>> (c) Don't believe everything you read in novels. >>> >>> Let's go with (b), and try to calculate the height D of the "arch" of the >>> horizon over the ruler's edge, and show it's too small to observe by the >>> naked eye. >>> >>> There's a diagram at: >>> >>> https://whitbywriters.wordpress.com/2018/02/26/the-curious- >>> incident-of-the-dead-flat-horizon >>> where I discuss the matter for non-mathematical readers. There I baldly >>> state that D is given by the formula: >>> >>> D = R * (1 – cos arcsin L/R) >>> >>> where L is the half-length of the straight-edge and R the radius of the >>> earth. >>> >>> Have I got it right? >>> >>> I calculated D = 1.4 nm, but omit to say how. (I used J of course). >>> >>> R=: 6378163 NB. equatorial radius of earth (m) >>> L=: 0.15 NB. half-length of metal ruler (m) >>> cos=: 2&o. >>> arcsin=: _1&o. >>> smoutput D=: R * (1 - cos arcsin L%R) >>> 1.41624e_9 >>> >>> Now 1.41624e_9 m is roughly 1.4 nm, less than the width of a DNA >>> molecule. >>> >>> But I suspect that my estimate of D is unreliable. Moreover it could be >>> out >>> by several orders of magnitude. Why? Because I'm calculating the arcsin >>> of >>> a very small number, and then calculating the cosine of the resulting >>> very >>> small angle, which may be pushing (o.) beyond its limits. >>> >>> Now I could use Pythagoras instead of trigonometry: >>> >>> D = R - √(R²-L²) >>> >>> which avoids using (o.) and may perform the entire calculation in >>> extended >>> precision without conversion to (float). I'll also work in nanometres >>> (nm), >>> not metres (m): >>> >>> R=: 6378163000000000x NB. (nm) >>> L=: 150000000x NB. (nm) >>> smoutput D=: R - %:(R*R)-(L*L) >>> 2 >>> >>> I thought I was getting 6-figure accuracy, so this result worries me. >>> 2 nm differs significantly from the result of the trig formula: 1.4 nm. >>> >>> Now (R*R)-(L*L) is extended precision, but (%:) is returning (float) not >>> (extended). Is it also corrupting the result? Is "2" just an artefact of >>> its algorithm? >>> >>> So let's work in picometres, to see if the "2" holds up under greater >>> precision: >>> >>> R=: 6378163000000000000x >>> L=: 150000000000x >>> smoutput D=: R - %:(R*R)-(L*L) >>> 2048 >>> >>> It does. This is encouraging. But (%:) still returns (float). >>> >>> Let's try different ways of calculating the square root of an extended >>> precision result: >>> >>> sq=: ^&2 >>> sqr=: %: >>> smoutput D=: R - sqr (sq R)-(sq L) >>> 2048 >>> sqr=: ^&0.5 >>> smoutput D=: R - sqr (sq R)-(sq L) >>> 2048 >>> sqr=: ^&1r2 >>> smoutput D=: R - sqr (sq R)-(sq L) >>> 3072 >>> …which is even more worrying! >>> >>> So which is the most dependable approximation to D? >>> 3 nm >>> 2 nm >>> 1.4 nm >>> none of the above? >>> >>> Is it robust enough to warrant my original deduction? >>> >>> Have I made a logical error somewhere? >>> ---------------------------------------------------------------------- >>> For information about J forums see http://www.jsoftware.com/forums.htm >>> >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm >> > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
