Length of arc of great circle?

[Guv]

Several months ago I helped somebody with formulae for Lunettes to be used to construct an approximately spherical object out of 20-30 strips of thin veneer.

Consider a series of equilateral polygons inscribed inside circles of constant latitude. Also consider equilateral polygons circumscribed around circles of constant latitude. If you imagine the shape generated by the polygons, it should approximate a sphere if the polygons have 20-30 (or more) sides.

I came up with the following formulae for polygons with N equal sides (not difficult to figure out).

InnerSide = 2*Radius*Sin(Pi / N)*Cos(Latitude)
OuterSide = 2*Radius*Tan(Pi / N)*Cos(Latitude)
Ark = 2*Radius*(Pi / N)*Cos(Latitude)

Ark is the arc of a circle of constant latitude. Any of the above formulae could be used to calculate widths for the desired Lunettes. It is my guess that the Ark formula is likely to be the best approximation of the three, but the other two formulae are guaranteed to result in developable surfaces. For those not familiar with the term, cylinders & cones are developable surfaces because local geometry on these surfaces is exactly equivalent to plane geometry. Such a surface can be straightened out and made into a piece of a plane without tearing or stretching it.

Does anybody know the formula for the arc of a great circle which connects two points at the same latitude?

For Latitude = 0 (the Equator), the above formula is correct, but what is the formula for other latitudes? I think that using such a formula might be more likely to result in a developable surface. In practice, I do not think it matters, but in theory, a developable surface is more correct.

[jim mcnamara]

For hams and pilots, "distance" of a great cricle bearing is the same as the length of the arc in degrees: what you want, I believe.

Formula returns cos(D) - distance in degrees of arc, C isn't relevant.

From the 1982 ARRL antenna book.

1. cos(D) = (sin(A) * sin(B)) + (cos(A) * cos(B) * cos(L))

2. cos(C) = (sin(B) - (sin(A) * cos(D))) / (cos(A) * sin(D))

WHERE:

A = YOUR latitude in degrees.
B = latitude of the other location in degrees.
L = YOUR longitude minus that of the other location. (Algebraic difference.)
D = Distance along path in degrees of arc.
C = True bearing from north if the value for sin(L) is positive. If
sin(L) is negative, true bearing is 360 - C.

To convert degrees of arc to statute miles, multiply by 69.041.
To convert degrees of arc to kilometers, multiply by 111.111.

[Guv]

Jim McNamara: Is it possible that the cos(D) formula is for half the angle between the points?

I experimented with MathCad7 and seemed to get the correct arc length when I used 2*Radius*D, where D is Inverse Cosine of the expression you posted for Cos(D).

[sail3005]

what exactly is a great circle again? i forgot.

[Guv]

Jim McNamara: There was an error in the work I did with MathCad7. The formulae you posted work like a charm. At least the values computed look correct. I have good upper and lower bounds for the geodesic (great circle) distances being computed, making your formulae seem right on the money.