Re: Formula for arc segment

"Steve B" <SurDO2Diver@xxxxxxxxxxx> wrote:
I have been getting a lot of help on this, and I appreciate it.

What I need to figure is street distance where the street is long and
curved. The street is part of an imaginary circle, and the two parallel
lines that make the street will be of slightly different lengths.

I want to measure in a straight line, the segment of the circumference of
the outer longest arc, then measure the height, then apply a formula that
will tell me the distance of the curved line.

I use aerial maps, therefore, I cannot always get center points, or some
of the other measurements, as they are none, or I could be only guessing,
and not exact as I need to be.

Some have suggested using a percentage of the circumference, but I'm
limited in what I can measure. But I can always measure the outer arc
length and the height.

Hope this helps. I know math pretty well, but because of a brain injury
a few years ago, have to now relearn things. I'm starting to get a grasp
on this, and know once I get it, it will stay there if only for a few
weeks.

I know trig and algebra. I need to read up some and understand radians,
and other terms used to try to give me the answer.

Thanks again, and I'm getting there.

Steve

The answer that lemel_man <binswood@xxxxxxxxxxx> posted yesterday looks
correct to me. That should give you all the math you need. Did you not
see it? Here it is:

lemel_man <binswood@xxxxxxxxxxx> wrote:
A straight line that joins the ends of a circular arc is called a chord.
If you take any 2 chords, AB and CD say, that cross at E (they don't
have to be at right angles to each other), then AE x EB = CE x ED.
In your case, AE = 506/2 = 253 = EB, CE = 27 and ED = 2r - 27 where r is
the radius of the circle.
Therefore 253x253 = 27x(2r-27), and, solving for r gives
r = (253x253+27x27)/2x27 = 1198.85
Because the chords are at right-angles, the angle subtended by half the
arc AB to the centre of the circle can be found by simple trig; its the
angle whose Sine is EB/r = 253/1198.85, which is 12.183 degrees, so the
angle subtended by the whole arc AB is twice this, or 12.183x2 = 24.366.
The ratio of this to 360 (degrees in whole circle) gives the ratio of
the arc length to the circumference of the circle. So the arc length is
2xPIx1198.85 x 24.366/360 = 509.83

Condensing all that down to a two formulas and simplifying a bit gives us:

If the straight line distance you measure is C (cord length),
and the height is H, then the formula is:

Diameter:
D = C*C/4H + H

Arc Length:
L = 2*PI*D*arcsin(C/D)/360

For your numbers:

C = 506
H = 27

D = 506*506/4*27 + 27 = 2397.70
L = 2*3.1415*2397.70*arcsin(506/2397.70)/360
L = 15064.7 * arcsin(0.21104)/360
L = 15064.7 * 12.183 / 360
L = 509.82

--
Curt Welch http://CurtWelch.Com/
curt@xxxxxxxx http://NewsReader.Com/
.

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