NCPMA
ADVANCED MAPPING
NORTH CAROLINA PROPERTY MAPPERS ASSOCIATION
ADVANCED MAPPING
SECTION 5-C
CURVES
5C-1
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5C.1 Curves
Straight (tangent) sections of roads are connected by curves in both the horizontal and
vertical planes. Curves used in horizontal planes to connect two straight tangent sections are
called horizontal curves. A simple curve is a circular arc connecting two tangents. A reverse
curve consists of two circular arcs tangent to each other, with their centers on opposite sides
of the alignment. There are several other circular curves. They are shown in Figure 5C-1.
Figure 5C-1
Circular curves.
5C.2 Definition of circular curve formulas
Circular curve components are shown in Figure 5C-2. Additional relationships used to
compute and plot curves are shown in Figure 5C-3.
5C-2
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Figure 5C-2
5C-3
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Elements of a Curve (for Figures 5C-2 and 5C-3)
P.C.
(Point of curvature): The point at which the curve begins.
T
(Tangent distance): A line that touches the curve at one point and is at a right angle
to the radius at the point of contact. Each curve has two tangents, always equal
length; the distance from the point of curvature to the point of intersection or from
the point of intersection to the point of tangency.
(Point of intersection): The point where the two tangents meet. The deflection
angle at this point is equal to delta.
(Deflection angle): Angle created between the continuation of one tangent and the
other tangent
P.I.
I
P.T.
(Point of tangency): The point at which the curve ends.
A.L.
(Arc length): The distance measured along the line of a curve from the P.C to the
P.T.
L.C.
R
(Long chord): Straight line from the P.C. to the P.T.
(Radius): The distance from the tangents to the center of curve or arc. Radii are
always perpendicular to the tangents at the point of curvature and the point of
tangency. The two Radii are always equal in length.
(Delta Angle): Also, know as the central angle. The angle made by the two radii
from the center of the arc to the point of curvature (P.C.) and point of tangency
(P.T).
Δ
P.R.C.
Point of Reverse Curvature): A point at which a curve in one- direction ends and
another curve in the opposite direction begins.
Back tangent – The last line drawn before the point of curvature (P.C.)
Forward tangent – The straight line, which continues when the curve ends at the (P.T.)
5C-4
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Figure 5C-3 Additional Curve Diagram Relationships
5C-5
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Section 5-C
Exercises
Exercise 5C-1
Plot at 1" = 100' the following simple curve.
Given Radius = 250'
Deflection angle at the PI = 40
Find chord length ____________________
Delta angle _________________________
Angle between tangent and long chord___
Tangent length ______________________
Exercise 5C-2
Plot at 1" = 100' the following simple curve.
Given tangent length 170 feet
Angle between tangent and long chord of 40
Find chord length ____________________
Delta angle _________________________
Radius ___________________________
5C-6
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ADVANCED MAPPING
Exercise 5C-3
Plot at 1" = 200' a curve to the left with a tangent length of 195 feet and a delta angle of 35
Exercise 5C-4
Plot at 1" = 200' a curve to the right with a tangent of 300 feet and a delta angle of 50
5C-7
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ADVANCED MAPPING
Exercise 5C-5
Given T=200’
Δ = 52°
R
∢ PI
∢ T & LC
LC
=
=
=
=
?
?
?
?
5C-8
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ADVANCED MAPPING
EXERCISE 5C-6
Plot the following reverse curve given the following:
Back tangent bearing of N 56°, East radius of the first curve of 475’ with a central angle of
93°, radius of the second curve of 750’ with a central angle of 44°, forward tangent of S 73°
E.
Note: Plot at 1”=200’
PC 
5C-9
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ADVANCED MAPPING
EXERCISE 5C-7
Plot the following reverse curve given the following:
Back tangent bearing of S 72° E; chord bearing of N 68° E 760’, chord bearing of N 58° E
688’, forward tangent bearing of N 86° E.
Note: Plot 1”=200
PC 
5C-10
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ADVANCED MAPPING
Exercise 5C-8
Plot the following description at 1” = 200’
Beginning at an iron pipe, thence proceed N 35° E, 710 feet to a nail in Harris Road; thence
along said Harris Road S 50° E 460’ to a point of curvature to the right; said curve has a
radius of 322 feet and a delta angle of 75° 30’; from the PT proceed S 28° W 640 feet to a
nail in said road; thence leaving said road, on an azimuth of 340°, 460 feet to a cherry tree;
thence with a deflection angle of 70° to the left proceed 250 feet to a painted rock; thence on
an angle to the left of 119 degrees 275 feet to the point of beginning.
POB
5C-11
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ADVANCED MAPPING
Exercise 5C-9
Plot at a scale of 1”=200’
To Wit:
All that part of Grantor’s land in Davis Township, Pirate County, North Carolina which lies
on both sides of and adjacent to the following described surveyed construction centerline of
State Route 66.
The surveyed construction of said Route 66 is described as:
Beginning at a point that lies 205 feet North 87 degrees East of the northwest corner of
property now or formerly owned by Billy Bob Oakes as shown in Deed Book 1219 Page 37
and recorded in the Pirate County Registry, said point is shown on the plans of the proposed
construction of Route 66 as station 2 + 35; thence along a curve to the right having a radius
of 450’, a central angle of 52° to station 6 + 43.93 which is a reverse curve to the left having
a radius of 700’, a central angle of 58° 30’ to station 13 + 59.71 which is a point of tangency;
thence north 80° 30’ east 300’ to station 16 + 59.71; 82°.
Widths of the right-of-way along the northerly side of the above described survey
construction centerline are as follows: A tract of land widening on a direct line from 40 feet
at station 0 + 00 to 60 feet at station 2 + 35; Thence remaining at 60 feet from station 2 + 35
to station 6 + 43.93; thence still remaining at 60’ until station 13 + 59.71; thence narrowing
in a direct line from 60 feet to 50 feet at station 16 + 59.71.
Widths of right-of-way along the southerly side of the above described survey construction
centerline of Route 66 are as follows;
A tract of land widening on a direct line from 30 feet at Station 0 + 00 to 50 feet at station 2 +
35; thence remaining at 50 feet until station 6 + 43.93; continuing at 50 feet until station 13 +
59.71; thence narrowing on a direct line from 50 feet to 40 feet at station 16 + 59.71.
5C-12
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ADVANCED MAPPING
Solutions
Section 5C
Exercise 5C-1
Plot at 1" = 100' the following simple curve.
Given Radius = 250'
Deflection angle at the PI = 40
Find chord length 170°
Delta angle 40°
Angle between tangent and long chord 20°
Tangent length 90’
Exercise 5C-2
Plot at 1" = 100' the following simple curve.
Given tangent length 170 feet
Angle between tangent and long chord of 40
Find chord length 260’
Delta angle 80°
Radius 200’
5C-13
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ADVANCED MAPPING
Solution 5C-3
Trig Calculation
Tan ½ Delta = opp/adj
Tan 17.5 = Tan/Rad
Tan 17.5 = 195/r
R= 195/Tan 17.5 = 618.46
5C-14
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Solution 5C-4
5C-15
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Solution 5C-5
Given T=200’
Δ= 52°
R
∢ PI
∢ T & LC
LC
=
=
=
=
404’
52°
26°
360’
5C-16
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ADVANCED MAPPING
Solution 5C-6
Plot the following reverse curve given the following:
Back tangent bearing of N 56°, East radius of the first curve of 475’ with a central angle of
93°, radius of the second curve of 750’ with a central angle of 44°, forward tangent of S 73°
E.
Note: Plot at 1”=200’
5C-17
NCPMA
ADVANCED MAPPING
Solution 5C-7
Plot the following reverse curve given the following:
Back tangent bearing of S 72° E; chord bearing of N 68° E 760’, chord bearing of N 58° E
688’, forward tangent bearing of N 86° E.
Note: Plot 1”=200
5C-18
NCPMA
ADVANCED MAPPING
Solution 5C-8
Plot the following description at 1” = 200’
Beginning at an iron pipe, thence proceed N 35° E, 710 feet to a nail in Harris Road; thence
along said Harris Road S 50° E 460’ to a point of curvature to the right; said curve has a
radius of 322 feet and a delta angle of 75° 30’; from the PT proceed S 28° W 640 feet to a
nail in said road; thence leaving said road, on an azimuth of 340°, 460 feet to a cherry tree;
thence with a deflection angle of 70° to the left proceed 250 feet to a painted rock; thence on
an angle to the left of 119 degrees 275 feet to the point of beginning.
5C-19
NCPMA
ADVANCED MAPPING
Solution 5C-9
Plot at a scale of 1”=200’
To Wit:
All that part of Grantor’s land in Davis Township, Pirate County, North Carolina which lies on both sides of and adjacent to the
following described surveyed construction centerline of State Route 66.
The surveyed construction of said Route 66 is described as:
Beginning at a point that lies 205 feet North 87 degrees East of the northwest corner of property now or formerly owned by Billy
Bob Oakes as shown in Deed Book 1219 Page 37 and recorded in the Pirate County Registry, said point is shown on the plans
of the proposed construction of Route 66 as station 2 + 35; thence along a curve to the right having a radius of 450’, a central
angle of 52° to station 6 + 43.93 which is a reverse curve to the left having a radius of 700’, a central angle of 58° 30’ to station
13 + 59.71 which is a point of tangency; thence north 80° 30’ east 300’ to station 16 + 59.71; 82°.
Widths of the right-of-way along the northerly side of the above described survey construction centerline are as follows: A tract
of land widening on a direct line from 40 feet at station 0 + 00 to 60 feet at station 2 + 35; Thence remaining at 60 feet from
station 2 + 35 to station 6 + 43.93; thence still remaining at 60’ until station 13 + 59.71; thence narrowing in a direct line from 60
feet to 50 feet at station 16 + 59.71.
Widths of right-of-way along the southerly side of the above described survey construction centerline of Route 66 are as follows;
A tract of land widening on a direct line from 30 feet at Station 0 + 00 to 50 feet at station 2 + 35; thence remaining at 50 feet
until station 6 + 43.93; continuing at 50 feet until station 13 + 59.71; thence narrowing on a direct line from 50 feet to 40 feet at
station 16 + 59.71.
5C-20