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Straight line walking

Re: Straight line walking

Postby litljortindan » Tue Sep 07, 2021 12:16 pm

litljortindan wrote:Surely in walking a chord length of 78.4km they will have deviated from a straight line by approximately
6371*(1-COS((ACOS((6371*6371*2-78.4*78.4)/(2*6371*6371))/2)))*1000 = 121m at the mid-point?


Or should that be (6371 - SQRT(6371*6371-POWER(6371*SIN(((78.5/40075)*2*PI())/2),2)))*1000 = 120m?
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Re: Straight line walking

Postby ChrisButch » Tue Sep 07, 2021 12:42 pm

Nicholas Crane's 'Two Degrees West' - a N-S walk on the 2% W meridian - is one of the more impressive (if eccentric) examples:

https://www.amazon.co.uk/Two-Degrees-West-English-Journey/dp/0140272364
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Re: Straight line walking

Postby litljortindan » Tue Sep 07, 2021 1:39 pm

litljortindan wrote:
litljortindan wrote:Surely in walking a chord length of 78.5km they will have deviated from a straight line by approximately
6371*(1-COS((ACOS((6371*6371*2-78.4*78.4)/(2*6371*6371))/2)))*1000 = 121m at the mid-point?


Or should that be (6371 - SQRT(6371*6371-POWER(6371*SIN(((78.5/40075)*2*PI())/2),2)))*1000 = 120m?


I noticed my error in taking the arc length of 78.5km as the chord length (which is actually 78.4km) so that should have been:

6371*(1-COS((ACOS((6371*6371*2-78.4*78.4)/(2*6371*6371))/2)))*1000 = 120.6m at the mid-point

or (6371 - SQRT(6371*6371-POWER(6371*SIN(((78.5/40075)*2*PI())/2),2)))*1000 = 120.3m
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Re: Straight line walking

Postby jmarkb » Tue Sep 07, 2021 3:16 pm

And how much difference is there in the horizontal between a great circle arc and a straight line on the OS national Grid projection?
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Re: Straight line walking

Postby simon-b » Tue Sep 07, 2021 3:27 pm

jmarkb wrote:And how much difference is there in the horizontal between a great circle arc and a straight line on the OS national Grid projection?

Maybe a map like this might help establish that?
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Re: Straight line walking

Postby litljortindan » Wed Sep 08, 2021 3:30 pm

jmarkb wrote:And how much difference is there in the horizontal between a great circle arc and a straight line on the OS national Grid projection?


https://www.ordnancesurvey.co.uk/documents/resources/guide-coordinate-systems-great-britain.pdf

https://pubs.er.usgs.gov/publication/pp1395

Answer's bound to be in there somewhere.
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Re: Straight line walking

Postby litljortindan » Wed Sep 08, 2021 3:38 pm

https://www.ordnancesurvey.co.uk/documents/resources/guide-coordinate-systems-great-britain.pdf

p10:
"heights in mainland Britain are measured relative to the tide-gauge in Newlyn, Cornwall, giving a reference surface which is about 80 centimetres below the Geoid."

It's a can of worms or possibly a rabbit hole.
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