GEFS (FOUNDAMENTALS OF SURVEYING)
CBLAMSIS UNIVERSITY OF THE CORDILLERAS REVERSED CURVES
REVERSED CURVE
Definition:
Reversed curve is a curve formed by two simple curves with their center of curvature
located on opposite side of their common tangent.
Types of Reversed Curve
1. Reversed curve with parallel tangents
2. Reversed curve with converging tangents
Reversed curve with parallel tangents
The azimuth or bearing of the back and forward tangents are the same.
X
V1
O2
T1 I1 P
PC
I1/2 R2 I2
R2
PRC
R1 I2/2
R1 PT
I1 I2 T2
V2
O1
Definition of elements:
PRC (Point of Reversed Curve) – the point where the two simple curves meet at the
common tangent.
P = perpendicular (shortest) distance between the parallel tangents
All other elements with subscript 1 and 2 refer to the elements of the first simple curve
and the second simple curve, respectively.
If the common tangent will intersect the back and forward tangents, which are parallel to
each other, the common tangent will make the same angle of intersection with the two
parallel tangents. Thus I1 = I2 = I.
̅̅̅̅̅̅̅̅̅̅̅
Also, the line 𝑃𝐶, 𝑃𝑅𝐶 and line ̅̅̅̅̅̅̅̅̅̅̅
𝑃𝑅𝐶, 𝑃𝑇 makes the same angle with the common tangent,
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
thus the line 𝑃𝐶, 𝑃𝑅𝐶, 𝑃𝑇 (long chord of the reversed curve) is a straight line.
The length of the common tangent is equal to the sum of the tangent distances of the two
simple curves. That is V1V2 = (T1 + T2).
Also, the long chord of the reversed curve (LC) is equal to the sum of the long chords of
the two simple curves. That is LC = (LC1 + LC2).
, GEFS (FOUNDAMENTALS OF SURVEYING)
CBLAMSIS UNIVERSITY OF THE CORDILLERAS REVERSED CURVES
Construct a line passing through V2 perpendicular to the two parallel tangents, right
triangle V2XV1 is formed. From the right triangle formed, P = (T1 + T2)(tan I).
Reversed Curve with Converging Tangents
The back and forward tangents of the reversed curve will intersect at the vertex, V.
PC T1 V1
I1 V
I
R1 O2
PRC R2
I1 I2
R1
O1 PT
I2
V2
Elements:
TL = tangent distance from point PC to point V (vertex)
TS = tangent distance from point PT to point V
I = angle of intersection of the back and forward tangents
All other elements with subscript 1 and 2 refer to the elements of the first simple curve
and the second simple curve, respectively.
Isolate triangle V1V2V
V1
I1 V
I
(180 – I2)
I2
V2
𝑆𝑢𝑚 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒𝑠 = (𝑛 − 2)1800
𝐼1 + (180 − 𝐼2 ) + 𝐼 = 180
𝐼2 = 𝐼 + 𝐼1
Sine Law:
𝑇𝐿 − 𝑇1 𝑇𝑆 + 𝑇2 𝑇1 + 𝑇2 𝑇𝐿 − 𝑇1
= = =
𝑠𝑖𝑛(180 − 𝐼2 ) 𝑠𝑖𝑛𝐼1 𝑠𝑖𝑛𝐼 𝑠𝑖𝑛𝐼2
(𝑇1 + 𝑇2 )𝑠𝑖𝑛𝐼2
CBLAMSIS UNIVERSITY OF THE CORDILLERAS REVERSED CURVES
REVERSED CURVE
Definition:
Reversed curve is a curve formed by two simple curves with their center of curvature
located on opposite side of their common tangent.
Types of Reversed Curve
1. Reversed curve with parallel tangents
2. Reversed curve with converging tangents
Reversed curve with parallel tangents
The azimuth or bearing of the back and forward tangents are the same.
X
V1
O2
T1 I1 P
PC
I1/2 R2 I2
R2
PRC
R1 I2/2
R1 PT
I1 I2 T2
V2
O1
Definition of elements:
PRC (Point of Reversed Curve) – the point where the two simple curves meet at the
common tangent.
P = perpendicular (shortest) distance between the parallel tangents
All other elements with subscript 1 and 2 refer to the elements of the first simple curve
and the second simple curve, respectively.
If the common tangent will intersect the back and forward tangents, which are parallel to
each other, the common tangent will make the same angle of intersection with the two
parallel tangents. Thus I1 = I2 = I.
̅̅̅̅̅̅̅̅̅̅̅
Also, the line 𝑃𝐶, 𝑃𝑅𝐶 and line ̅̅̅̅̅̅̅̅̅̅̅
𝑃𝑅𝐶, 𝑃𝑇 makes the same angle with the common tangent,
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
thus the line 𝑃𝐶, 𝑃𝑅𝐶, 𝑃𝑇 (long chord of the reversed curve) is a straight line.
The length of the common tangent is equal to the sum of the tangent distances of the two
simple curves. That is V1V2 = (T1 + T2).
Also, the long chord of the reversed curve (LC) is equal to the sum of the long chords of
the two simple curves. That is LC = (LC1 + LC2).
, GEFS (FOUNDAMENTALS OF SURVEYING)
CBLAMSIS UNIVERSITY OF THE CORDILLERAS REVERSED CURVES
Construct a line passing through V2 perpendicular to the two parallel tangents, right
triangle V2XV1 is formed. From the right triangle formed, P = (T1 + T2)(tan I).
Reversed Curve with Converging Tangents
The back and forward tangents of the reversed curve will intersect at the vertex, V.
PC T1 V1
I1 V
I
R1 O2
PRC R2
I1 I2
R1
O1 PT
I2
V2
Elements:
TL = tangent distance from point PC to point V (vertex)
TS = tangent distance from point PT to point V
I = angle of intersection of the back and forward tangents
All other elements with subscript 1 and 2 refer to the elements of the first simple curve
and the second simple curve, respectively.
Isolate triangle V1V2V
V1
I1 V
I
(180 – I2)
I2
V2
𝑆𝑢𝑚 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒𝑠 = (𝑛 − 2)1800
𝐼1 + (180 − 𝐼2 ) + 𝐼 = 180
𝐼2 = 𝐼 + 𝐼1
Sine Law:
𝑇𝐿 − 𝑇1 𝑇𝑆 + 𝑇2 𝑇1 + 𝑇2 𝑇𝐿 − 𝑇1
= = =
𝑠𝑖𝑛(180 − 𝐼2 ) 𝑠𝑖𝑛𝐼1 𝑠𝑖𝑛𝐼 𝑠𝑖𝑛𝐼2
(𝑇1 + 𝑇2 )𝑠𝑖𝑛𝐼2
