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This chapter explains several curve and tangent line-curve fitting situations.  They can be used with other COGO tools to construct complex curvilinear traverses having to fit specific mathematical conditions.

A. Fitting arc through three points

Three non-linear points define a circular arc, Figure I-1

Figure I-1
Three point arc

If the coordinates of the three points (1, 2, and 3) are known, the arc radius (R) and radius point (O) coordinates can be determined.

Equations I-1 and I-2 are used to compute the radius point coordinates.

Equation I-1
Equation I-2

The coefficients for Equation I-1 are:

Equation I-3
Equation I-4
Equation I-5
Equation I-6


Once the radius point coordinates are determined, the arc radius can be computed from Equation I-7 .

Equation I-7

Point i is any of the three points used to define the arc.

 


B. Tangent line-arc intersection

The next two solutions deal with tangent a tangent line and arc.

1. Radius from line direction and radius point

Known information

Determine

Figure I-2
Radius determination

Solution process

Inverse along the line from point Q to the radius point, O, to determine the direction and length of the line.

Compute angle Q as the difference between the directions AzQO and AzQP, subtracting the smaller from the larger, depending on which side of line QO line QP is.

Solve the following equations

Equation I-8
Equation I-9
Equation I-10
Equation I-11

2. Line direction from radius point and radius

Known information

Determine

Determine the direction of a line tangent to an arc.

Figure I-3
Tangent direction determination

Solution process

Inverse along the line from point point Q to the radius point O to determine its direction and length.

Compute angle g using Equation I-12.

Equation I-12

Angle g is used with the direction of line point Q to radius point O to determine the direction from point Q to each intersection point.

The distance from point Q to each intersection point is:

Equation I-13

To compute the coordinates of P1, perform a Forward Computation from point Q:

Equation I-13
Equation I-14
Equation I-15

To compute the coordinates of point P2, perform a Forward Computation from  point Q:

Equation I-16
Equation I-17
Equation I-18

 


C. Three-tangent arc

Three non-colinear intersecting lines can serve as tangent lines to define a circle, Figure I-5.

 

Figure I-5
Tangent to three lines

Figure I-6 shows the primary geometry of the three-tangent arc.  

Figure I-6
Geometry - curve to left

Figure I-7 shows geometry for a curver to the right, as defined by the tangent azimuths.

Figure I-7
Geometry - curve to the right

Points I, J, and K are the tangent points. These divide the curve into two tangent arcs having the same radius with points G and H as their Points of Intersection (PI).

Solution process

Determine PIs, G and H, by direction-direction intersection. Inverse between them to obtain the distance, dGH.

Using the azimuths, compute deflection angles ΔG and ΔH at the PIs.

Compute curve radius using Equation I-19

Equation I-19

R is positive for a curve to the right; negative for a curve to the left.

Distances dGJ and dGI are the tangent distances for the first curve; distances dHK and dHJ are the tangent distances for the second curve. To compute an arc's tangent distance use:

Equation I-20

To compute the coordinates of the radius point

The coordinates of the radius point can also be computed by Forward Computation sequences:

 


D. Examples

1. Three point curve

Three known coordinate points are

Point North (ft) East (ft)
A 902.72 1751.91
B 870.76 1237.76
C 538.57 1293.16

Determine the radius of the circle passing through the points and coordinates of the radius point.

Sketch

Compute the coefficients using Equations I-3 through I-6

Use the coefficients in Equations I-1 and I-2 to get radius point coordinates

Get radius from Equation I-7

Using point A

Check using point B

Checks within rounding.

Answers

Radius   294.42'
Radius pt coords   744.39' N
    1503.68' E

2. Tangent line

What is the direction of the line line tangent to the arc shown below? What are the coordinates of the intersection point?

Inverse from point W to the radius point O

Using Equation I-12 compute the angle g between the line to the radius point and tangent line.

Determine the azimuth and length of the tangent line

.

Since the tangent line is left of the line to the radius point, to get its azimuth subtract g from AzWO' Equation, I-16.

Its length comes from Equation I-13

Forward Computation, Equations I-17 and I-18, to compute tangent point coordinates

Answers

Tangent azimuth   72°45'42"
Tangent pt coords   1501.00' N
    1810.34' E

3. Three-tangent arc

What is the radius of the arc that is tangent to the three line shown?

Label the geometry

Establsh points G and H using direction-direction intersections (results shown, comps left to user)

Point North East
G 1849.672 1305.139
H 1735.850 1670.923

ΔG and ΔH from the azimuths (comps left to user)

ΔG 63°05'03"
ΔH 49°03'40"

Inverse Computation to obtain distance from point G to point H  (comps left to user): 383.084

Apply Equation I-19 to get the arc radius

Tangent distance for first arc section (comps left to user): 219.721

Radius point coordinates (comps left to user): 1,442.592' N and 1,408.580' E

Answers

Radius   357.96'
Radius pt coords   1,442.59' N
    1,408.58' E
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