SME
Surveying · Problem type
Linear misclosure and relative precision
Use this for “A traverse 1250 m round sums to Σlat = −0.045 m and Σdep = +0.060 m. Find e, its direction and the precision.”
Diagram
e
=
√
Σlat² + Σdep²
=
√
−
0.045
2
+
0.06
2
=
0.075
m;
precision
one
part
in
P
e
=
16,700.
The gap is drawn 892 times the traverse’s scale.
This is an example. Type your own values to start a new problem.
e (m)
=
(
Σlat (m)
)
2
+
(
Σdep (m)
)
2
x
=
P (m)
÷
e (m)
βₑ
Direction of the gap
worked out (not typed)
Formulas
e
=
√
Σlat² + Σdep²
0.075
=
√(
(−0.045)
2
+
0.06
2
)
x
=
P
e
16,667
=
1,250
0.075
βₑ = atan2(Σdep, Σlat)
S 53°07′48″ E = atan2(0.06, −0.045)
Assumptions
•
Σlat and Σdep are the measured courses’ sums; a perfect closed traverse sums to 0 both ways.
•
The precision is one part in x: one metre of gap in x metres of traverse.
•
One part in 10,000 is a common standard for property surveys.
Step-by-step
WE KNOW
Σlat = −0.045 m, Σdep = 0.06 m, P = 1,250 m
FIND
linear misclosure (e), precision, one part in x (x), direction of the gap (βₑ)
Step 1 · Find linear misclosure (e)
Use e = √(Σlat² + Σdep²)
The
gap’s
north
and
east
parts
are
the
legs
of
a
right
triangle;
e
is
its
hypotenuse.
e
=
√(
(−0.045)
2
+
0.06
2
)
e
=
√
0.002025 + 0.0036
e
=
√
0.005625
e
=
0.075
m
Step 2 · Find precision, one part in x (x)
Use x = P ÷ e
How many metres of traverse for each metre of gap.
x
=
1,250
0.075
x
=
16,667
Step 3 · Find direction of the gap (βₑ)
Use βₑ = atan2(Σdep, Σlat)
The way the traverse fails to close: tan⁻¹ of Σdep over Σlat, turned into its quarter.
βₑ = atan2(0.06, −0.045)
(east, north) = (0.06, −0.045) points south-east: add 180°
azimuth
=
tan⁻¹(
0.06
−0.045
)
+
180°
=
−53°07′48″
+
180°
=
126°52′12″
126°52′12″ = S 53°07′48″ E
βₑ = S 53°07′48″ E
Check
Put the numbers back in. Both sides should match.
0.075
=
√(
(−0.045)
2
+
0.06
2
)
✓
16,667
=
1,250
0.075
✓
126°52′12″
=
tan⁻¹(
0.06
−0.045
)
+
180°
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Traverse computations
Main lesson
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