E4004 Surveying Computations A Area Problems. To Cut Off an Area by a Line Passing Through a Fixed...
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Transcript of E4004 Surveying Computations A Area Problems. To Cut Off an Area by a Line Passing Through a Fixed...
E4004 Surveying Computations A
Area Problems
To Cut Off an Area by a Line Passing Through a Fixed Point• The bearing and distance BP is known
B
P
X
BrgDist
Brg
• The bearing BX is known• The required area BPX is known
A
• Calculate the bearing and distance PX and the distance BX
To Cut Off an Area by a Line Passing Through a Fixed Point• The angle at B is determined from
the bearing difference
B
P
X
BrgDist
Brg
• The general formula for the area of a triangle is
A
CabABC sin2
1
PBXBXBPPBX sin*2
1
PBXBP
PBXBX
sin
*2
C
B
A
a
b
To Cut Off an Area by a Line Passing Through a Fixed Point
• The bearing & distance of the line PX can be calculated by closing PBX
B
P
X
BrgDist
Brg
• Also check that the area PBX calculates to the correct area by using the CLOSE programA
PBXBP
PBXBX
sin
*2
To Cut Off an Area by a Line Passing Through a Particular Point• A farmer wants to fence off a particular
area from a large paddock.There is an existing trough which must be accessible to stock on both sides of the new fence.
To Cut Off an Area by a Line Passing Through a Particular Point
• The bearings of BC and BD are known.
B
C
D
Brg
Brg
• The bearing and distance BP can be measured.
BrgDist
P
• The required area is A
A
To Cut Off an Area by a Line Passing Through a Particular Point
• Note that there will be two solutions
B
C
D
Brg
Brg
BrgDist
P
• Such that
C’
D’
'" PDDAreaPCCArea
A
To Cut Off an Area by a Line Passing Through a Particular Point
B
C
D
Brg
Brg
BrgDist
P
x
y
A
• Let
CBP PBD
yBD xBC
To Cut Off an Area by a Line Passing Through a Particular Point
B
C
D
Brg
Brg
BrgDist
P
x
y
sin**2
1BPxBCP
sin**2
1BPyBPD
sin**2
1sin**
2
1BPyBPxA
yx
BCD
yxBCD
yxBCD
sin*
*2
sin***2
sin**2
1
A
To Cut Off an Area by a Line Passing Through a Particular Point
B
C
D
Brg
Brg
BrgDist
P
x
y
sin**2
1sin**
2
1BPyBPxA y
x
A
sin*
*2
A
sin**sin*
*2*
2
1sin**
2
1BP
x
ABPxA
sin** xA
sin***sin*sin**2
1*sin* 2 BPAxBPxA
Multiply both sides of the
equation by, x sin()
Re-write in terms of x
sin**sin**2
1xBPx sin**BPA
To Cut Off an Area by a Line Passing Through a Particular Point
B
C
D
Brg
Brg
BrgDist
P
x
y
A
sin***sin*sin**2
1*sin* 2 BPAxBPxA
sin*sinsinsin2
10 2 BPAxAxBP
)sin(sin2
1 BPa
)sin( AbsinBPAc
This equation is in quadratic form and can be solved for x
Make the LHS equal zero
To Cut Off an Area by a Line Passing Through a Particular Point
• Write a program to solve for x in a quadratic given values for a, b and c
• OR write a solver program which will solve for x, a, b or c
02 cbxaxbc
acbbx
2
42
To Cut Off an Area by a Line Passing Through a Particular Point When the Figure is not a Triangle
• It is required to cut off a given area CQRSTD by a line passing through P
C
Q
R
S
T
D
P
• The bearings and distances QR, RS and ST are known whilst the position of P has been located from Q
Brg & Dist
Brg & Dist
• Only the bearings are known for CQ and TD
Brg
Brg
A
To Cut Off an Area by a Line Passing Through a Particular Point When the Figure is not a Triangle
• Extend CQ and DT to intersect at B
C
Q
R
S
T
D
P
• The figure CBDF is the same as that formed in the earlier example provided the required area is made equal to the sum of Area QRSTB and A
Brg & Dist
Brg & Dist
Brg
Brg
B
A
• The dimensions of lines TB and BQ can be calculated by closing QRSTB and the line BP by closing BQP