CIVL1810 Chap.11 Angles, Zenith Distances and Trigonometric Heighting
Angles, Zenith Distances and Trigonometric Heighting
This is the multi part computation the paper opens with, and it is stated the same way in successive years: the reduced level of a spire is required, the top is intersected from two points using horizontal and zenith angles, the horizontal distance to the base is not known, and independent computations are made from each station and meaned.
The marks are distributed across small parts, and the first four of them are reductions of raw field readings rather than trigonometry. Every angle is observed on both faces so that instrumental errors equal and opposite on the two faces cancel when the readings are meaned.
What this chapter covers
- 01
Why the horizontal distance to an inaccessible point has to be solved rather than measured
- 02
Face left and face right observation, and what meaning the two faces removes
- 03
Reducing a horizontal angle: subtract on each face first, then mean the two angles
- 04
Why a circle reading is a direction rather than an angle
- 05
Zenith angles measured down from the vertical, so a horizontal sight reads 90 degrees
- 06
The two face readings summing to 360 degrees for a perfect instrument
- 07
The vertical index error as half the departure from 360 degrees
- 08
Correcting a zenith angle from the two face readings
- 09
The third angle of the intersection triangle from the angle sum
- 10
The sine rule pairing each side with the angle opposite it
- 11
Height difference as horizontal distance multiplied by the cotangent of the zenith angle
- 12
Adding the station reduced level and the instrument height to obtain the target level
- 13
Two independent computations, their agreement as the check, and the mean as the reported result
Reducing raw observations before any trigonometry
- +1Reduce each face to an angle by subtraction before meaning anything. Face left gives 41 degrees 15 minutes 00 seconds less 00 degrees 00 minutes 20 seconds, which is 41 degrees 14 minutes 40 seconds. Face right gives 221 degrees 15 minutes 40 seconds less 180 degrees 00 minutes 40 seconds, which is 41 degrees 15 minutes 00 seconds.
- +1Mean the two reduced angles. Half of the sum of 41 degrees 14 minutes 40 seconds and 41 degrees 15 minutes 00 seconds is 41 degrees 14 minutes 50 seconds, which is the mean horizontal angle BAC.
- +1Handle the zenith angle by summing the faces first. 86 degrees 40 minutes 30 seconds plus 273 degrees 19 minutes 50 seconds is 360 degrees 00 minutes 20 seconds, so the index error is plus 10 seconds and the corrected zenith angle is 86 degrees 40 minutes 30 seconds less 10 seconds, which is 86 degrees 40 minutes 20 seconds. Being below 90 degrees, the sight is elevated and the target lies above the trunnion axis.
Key terms
- Face left and face right
- Two observations of the same angle with the telescope transited and the instrument turned through 180 degrees. Instrumental errors that are equal and opposite on the two faces cancel when the reduced angles are meaned.
- Horizontal circle reading
- A direction relative to an arbitrary circle orientation, not an angle. An angle exists only as the difference between two directions observed from the same setup on the same face.
- Zenith angle
- A vertical angle measured down from the vertical, so a horizontal sight reads 90 degrees. Values below 90 degrees indicate an elevated sight and values above indicate a depressed one.
- Vertical index error
- Half the amount by which the two face readings of a zenith angle depart from 360 degrees. It is subtracted from the face left reading to obtain the corrected angle.
- Intersection
- Fixing an inaccessible point by observing it from both ends of a measured baseline. Two angles and one measured length determine the triangle, so the two unknown distances follow from the sine rule.
- Trigonometric heighting
- Obtaining a height difference from a horizontal distance and a vertical angle. With a zenith angle the multiplier is the cotangent, and the station reduced level and instrument height must both be added to reach the target level.
Angles, Zenith Distances and Trigonometric Heighting FAQ
Why can the two faces not simply be averaged as raw readings?
Because on face right the circle reads roughly 180 degrees away from its face left counterpart, so the average of the raw readings has no physical meaning. The correct order is to reduce each face to an angle by subtraction, then average the two angles. If the two reduced angles differ by more than the instrument's expected spread, the observation is suspect and should be repeated rather than averaged.
Why is the multiplier the cotangent rather than the tangent?
Because a zenith angle is measured from the vertical rather than from the horizontal. A nearly horizontal sight has a zenith angle close to 90 degrees and therefore a very large tangent and a very small cotangent, which correctly gives a small height difference. Using the tangent inverts the relationship, and for a nearly horizontal sight it inverts it enormously.
What is the point of computing the level twice?
The two computations share only the measured baseline, so if they agree to a few millimetres then the angles at both ends and both zenith reductions have been done correctly. That agreement is the check, and it is why the task asks for independent computations and a mean rather than a single answer. Reporting only the mean throws away the reason for doing it twice.
Which mistakes cause the two independent results to disagree?
Most often a zenith angle used with the tangent instead of the cotangent, or a sine rule pairing swapped, since the side you solve for must be opposite the angle you use.
Beyond those, an error in either instrument height shifts one result by about a metre and a half, an uncorrected index error at one station only biases that station's height, and over long sights curvature and refraction affect the two lines of sight by different amounts.
Exam move
Work this chapter in the order the paper does, because that order protects you. Reduce the raw readings first and completely: mean each horizontal angle, sum each pair of zenith faces to expose the index error, and only then start any trigonometry.
Write the four relations you need on one card, namely the angle sum, the sine rule, the cotangent relation and the reduced level assembly, and drill converting between degrees, minutes and seconds and decimal degrees without mixing them mid calculation.
Practise on your own invented triangles where you know the answer, and check your work the way the question does, by computing the target level twice from different ends and comparing the two.
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