AREA 10 Chord Spiral

Formulas

Formulas for the Exact Determination of the Functions of the Ten-Chord Spiral When the Central Angle Does Not Exceed 45 Degrees

 
 
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10) R = 50 / sin ( 1/2 D )
(11) Z = X - R sin 
(12) o = Y - R vers 
(13) Ts = (R + o) tan( 1/2 I ) + Z
(14) Es = (R + o) exsec( 1/2 I ) + o
 

Formulas for Field Use

The formulas presented above are best adapted for the preparation of tables.  For use in the field, the following empirical formulas are sufficiently accurate and have the advantage that they do not require the computation of the long chord.  The formulas can all be applied for the functions of any parts of the spiral without serious error, thought they are derived for the completed spiral.
 
(15) a=1/3  
A=1/3 
(16)  a=10 ks minutes 
A=10 kS minutes
Formulas (15) and (16) are sufficiently accurate for turning deflection when  (or ) does not exceed 15 degrees.

A similar approximation may be used when the transit is set at an intermediate point on the spiral if the included central angle from the transit point to the point of sight, less the included angle from the T.S. to the transit point, does not exceed 15 degrees.
 
(17) X=L - L(1/3 vers 3/4  + 1/22 vers 1/2 )
(18) Y=(L/39) (20 sin 1/2  + 3 sin )
(19) U=L (2/3 + 10/39 exsec 1/2  + 1/10 vers 1/4  )
(20) V=L (1/3 + 10/39 exsec 1/2 )
(21) o=L/10 (sin 1/2  + sin 1/3 ) cos 1/2 D
(22) Z=L (0.5 - .12885 vers 1/2 ) - .073 D sin 
(23) L=(370.82 / (cos 21/60 D)) (1 + .000018 Do) sqrt o/D
 

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