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March 17th, 2017, 12:45 AM  #1 
Newbie Joined: Jul 2016 From: Ringwood Posts: 7 Thanks: 0  Can anyone help me with a problem I'm having working out an arc
Hi I'm having problems drawing an arc of drawn a bit of a basic sketch if worked out I need my arc to have a rad of 5646mm but can anyone tell me what the height of line A will be and even shown me a formula please David 
March 17th, 2017, 04:42 AM  #2 
Senior Member Joined: Jun 2015 From: England Posts: 675 Thanks: 194 
I don't know what accuracy you require or if you know any trigonometry. The simplest way of setting this out is to use the formula that the deviation of a circular curve from the straight (tangent) is almost exactly the square of the length measured along the tangent divided by twice the radius. So if you set up a tangent, parallel to the ground, at the midpoint, which is also the lowest point, Then you can mark off any point on the curve by adding the offset to 1220. You have to work from the centre for this. I note that by the time you reach the extremity the calculated offset is 296, but you show 310. If this small error is unacceptable (it is only this much at the extremity) then we will need to look for a more accurate, but more complicated, trigonometric formula. 
March 17th, 2017, 02:45 PM  #3 
Newbie Joined: Jul 2016 From: Ringwood Posts: 7 Thanks: 0 
Thanks for the help just struggling to get the formula working on my calculator lol

March 17th, 2017, 06:35 PM  #4 
Math Team Joined: Jul 2011 From: Texas Posts: 2,639 Thanks: 1317  
March 17th, 2017, 07:51 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 18,034 Thanks: 1393  For radius r and length along the tangent x, where x $\small\leqslant$ r, the deviation (by Pythagoras) = r  √(r²  x²), which is 304.8001... for r = 5646 and x = 1830.

March 17th, 2017, 09:33 PM  #6 
Newbie Joined: Jul 2016 From: Ringwood Posts: 7 Thanks: 0 
Thanks skipjack a formula I can read, thankyou so much Last edited by Nutcase1; March 17th, 2017 at 09:35 PM. 
March 18th, 2017, 01:43 AM  #7 
Senior Member Joined: Jun 2015 From: England Posts: 675 Thanks: 194 
In quoting me and then posting this Are you disagreeing with my formula or supplying one of the many more accurate and more difficult formulae I mentioned? 
March 18th, 2017, 06:23 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 18,034 Thanks: 1393 
Trigonometry isn't needed. Any exact formula could be simplified to the exact formula I gave, which can be expanded to justify the approximate formula you gave.

March 18th, 2017, 07:49 AM  #9 
Senior Member Joined: Jun 2015 From: England Posts: 675 Thanks: 194  So are you saying that surveyors, mining and civil engineers were using the wrong formula for the last couple of hundred years?

March 18th, 2017, 11:52 PM  #10 
Global Moderator Joined: Dec 2006 Posts: 18,034 Thanks: 1393 
That depends on what you mean by "wrong". r  √(r²  x²) = r((x/r)²/2 + (x/r)$^4$/8 + (x/r)$^6$/16 + . . . ), so the approximate formula may be close enough to the exact value if x/r is small, but the exact formula is quite easy to evaluate. 

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