Laying Off Angles in Surveying
Case 1
The
angle is to be laid out no more precisely than the least count of the
theodolite or total
station. Assume that you must complete a
route survey where a deflection angle (31°12_ R)
has been determined from aerial photos to
position the cL (centerline) clear of natural obstructions [Figure
4.12(a)].
The
surveyor sets the instrument at the PI (point of intersection of the tangents),
and then sights the back line with the telescope reversed and the horizontal
circle set to zero.
Next,
the surveyor transits (plunges) the telescope, turns off the required
deflection
angle, and sets a point on line.
The
deflection angle is 31°12_ R and a
point is set at a_. The
surveyor then loosens the lower motion (repeating instruments), sights again at
the back line, transits the telescope, and turns off the required value (31°12_ _ 2
_ 62°24_).
It is very likely that this line of sight will
not precisely match the first sighting at a_; if the
line does not cross a_, a new mark
a_ is
made and the instrument operator gives a sighting on the correct point a, which
is midway between a_ and a_.
Case 2
The
angle is to be laid out more precisely than the least count of the instrument
permits.
Assume that an angle of 27°30_45_ is
required in a heavy construction layout, and that a
01-minute theodolite is being used.
In Figure 4.12(b), the theodolite is set up at
A zeroed on B with an angle of 27°31_ turned to
set point C_. The angle is then repeated to point C_
a suitable number of times so that an accurate value of that angle can be
determined.
Let’s assume
that the scale reading after four repetitions is 110°05_,
giving a mean angle value of 27°31_15_ for
angle BAC_. If the layout distance of AC is 250.00 ft,
point C can be located precisely by measuring from C_ a
distance C_C: C_C _ 250.00
tan 0°00_30_
_ 0.036
ft
After point C has been located, its position
can be verified by repeating angles to it from point B [Figure 4.12(b)].
Prolonging a Straight Line (Double Centering)
Prolonging
a straight line (also known as double
centering)
is a common survey procedure used every time a straight line must be prolonged.
The
best example of this requirement is in route surveying, where straight lines
are routinely prolonged over long distances and often over very difficult
terrain. The technique of reversion (the same technique as used in repeating,
or doubling, angles) is used to ensure that the straight line is prolonged
properly.
In
Figure 4.13, the straight line BA is to be prolonged to C [see also Figure
4.10(a)].
With the instrument at A, a sight is made
carefully on station B. The telescope is transited,
and a temporary point is set at C1.
The
theodolite is revolved back to station B (the telescope is in a position
reversed to the original sighting), and a new sighting is made. The telescope
is transited, and a temporary mark is made at C2, adjacent to C.
(Note: Over short
distances, well-adjusted theodolites will show no appreciable displacement
between points C1 and C2. However, over the longer distances normally
encountered in this type of work, all theodolites and total stations will
display a displacement between direct and reversed sightings.
(The longer the forward sighting, the greater
the displacement.) The correct location of station C is established midway
between C1 and C2 by measuring with a steel tape.
Bucking-In (Interlining)
It
is sometimes necessary to establish a straight line between two points that themselves
are not intervisible (i.e., a theodolite or
total station set up at one point cannot, because of
an intervening hill, be sighted at the other
required point).
It is
usually possible to find an intermediate position from which both points can be
seen.
In Figure 4.14, points A and B are not
intervisible, but point C is in an area from which both A and B can be seen.
The bucking-in procedure is as follows.
The instrument is set up in the area of C (at
C1) and as close to line AB as is possible to estimate. The instrument is
roughly leveled and a sight is taken on point A; then the telescope is
transited and a sight is taken toward B.
The
line of sight will, of course, not be on B but on point B1 some distance away.
Noting roughly the distance B1B and the position of the instrument between A
and B (e.g., halfway, one-third, or one-quarter of the distance AB), the
surveyor estimates proportionately how far the instrument is to be moved to be
on the line AB.
The
instrument is once again roughly leveled (position C2), and the sighting
procedure is repeated. This trial-and-error technique is repeated until, after
sighting A, the transited line of sight falls on point B or close enough to
point B so that it can be set precisely by shifting
the theodolite on the leveling head shifting
plate.
When
the line has been established, a point is set at or near point C so that the
position can be saved for future use. The entire procedure of bucking-in can be
accomplished in a surprisingly short period of time.
All but the final instrument setups are only
roughly leveled, and at no time does the instrument have to be set up over a
point.
Intersection
of Two Straight Lines
The
intersection of two straight lines is also a very common survey operation. In
municipal surveying, street surveys usually begin (0 _ 00) at the
intersection of the centerlines cL of two streets, and the station and angle of
the intersections of all subsequent streets’ cL are routinely determined. Figure 4.15(a)
illustrates the need for intersecting points on a municipal survey, and Figure
4.15(b) illustrates how the intersection point is located.
In
Figure 4.15(b), with the instrument set on a Main Street station and a sight
taken also
on the Main Street cL (the longer
the sight, the more precise the sighting), two points
(2–4 ft apart) are established on the Main
Street cL, one point
on either side of where the
surveyor estimates that the 2nd Avenue cL will
intersect.
The
instrument is then moved to a 2nd Avenue station, and a sight is taken some
distance away, on the far side of the Main Street cL.
The
surveyor can stretch a plumb bob string over the two points (A and B) established
on the Main Street cL, and the
instrument operator can note where on the string the 2nd Avenue cL intersects.
If the two points (A and B) are reasonably close
together (2–3 ft), the surveyor can use the
plumb bob itself to take a line from the instrument operator on the plumb bob
string; otherwise, the instrument operator can take a line with a pencil or any
other suitable sighting target. The intersection point is then suitably marked
(e.g., nail and flagging on asphalt, wood stake with tack on ground), and then
the angle of intersection and the station (chainage) of the point can be
determined. After marking and referencing the intersection point, the surveyors
remove the temporary
markers A and B.
Prolonging a
Measured Line by Triangulation
over an
Obstacle
In
route surveying, obstacles such as rivers or chasms must be traversed. Whereas
double
centering can conveniently prolong the
alignment, the station (chainage) may be deduced
from the construction of a geometric figure.
In Figure 4.16(a), the distance from 1 _ 121.271 to
the station established on the far side of the river can be determined by
solving the constructed triangle (triangulation). The ideal (geometrically
strongest—see Section 9.8)
Triangle
is one with angles close to 60° (equilateral), although angles as small as 20°
may be acceptable. Rugged terrain and heavy tree cover adjacent to the river
often result in a less than optimal geometric figure. The baseline and a
minimum of two angles are measured so that the missing distance can be
calculated. The third angle (on the far side of the river) should also be
measured to check for mistakes and to reduce errors.
See
Figure 4.16(b) for another illustration of triangulation. In Figure 4.16(b),
the line
must be prolonged past an obstacle: a large
tree. In this case, a triangle is constructed with the three angles and two
distances measured, as shown.
As
we noted earlier, the closer the constructed triangle is to equilateral, the
stronger is the calculated distance (BD).
Also, the optimal equilateral figure cannot always be constructed due to topographic constraints, and angles as small as 20° are acceptable for many surveys.
In
property surveying, obstacles such as trees often block the path of the survey.
In route
surveying, it is customary for the surveyor
to cut down the offending trees (later construction will require them to be
removed in any case), but in property surveying, the property owner would be
quite upset to find valuable trees destroyed just so the surveyor could establish
a boundary line.
Thus,
the surveyor must find an alternative method of providing distances and/or locations
for blocked survey lines.
Figure
4.17(a) illustrates the technique of right-angle offset. Boundary line AF
cannot be run because of the wooded area. The survey continues normally to
point B, just clear of
the wooded area. At B, a right angle is
turned (and doubled), and point C is located a sufficient distance away from B
to provide a clear parallel line to the boundary line.
The instrument is set at C and sighted at B
(great care must be exercised because of the short sighting distance); an angle
of 90° is turned to locate point D. Point E is located on the boundary line using
a right angle and the offset distance used for BC. The survey can then continue
to F.
If distance
CD is measured, then the required boundary distance (AF) is AB _ CD
_ EF.
If
intermediate points are required on the
boundary line between B and E (e.g., fencing
layout), a right angle can be turned from a
convenient location on CD, and the offset
distance (BC) is used to measure back to the
boundary line.
Use of this technique minimizes the
destruction of trees and other obstructions.
In Figure 4.17(b), trees are scattered over
the area, preventing the establishment
of a right-angle offset line. In this case, a
random line (open traverse) is run (by deflection
angles) through the scattered trees.
The
distance AF is the sum of AB, EF, and
the resultant of BE. The technique of right-angle offsets has a
larger potential for error because of the weaknesses associated with several
short sightings; at the same time, however, this technique gives a simple and
direct method for establishing intermediate points on the boundary line.
By contrast,
the random-line and triangulation methods provide for stronger geometric
solutions to the missing property line distances, but they also require less
direct
and much more cumbersome calculations for the
placement of intermediate line points
(e.g., fence layout).
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