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

• Coordinates at point Q
• Direction from point Q
• Coordinates of radius point, point O

Determine

• Radius of tangent arc
• Coordinates of tangent point P
 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

• Coordinates of radius point, point O
• Arc radius
• Coordinates of point Q

Determine

• Direction of a line from point Q tangent to the arc; there are two possible directions, compute both
• Coordinates of the two tangent points P1 and P2

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

• Forward Computation from point G to point J using AzB and tangent distance dGJ
• Compute azimuth to the radius point,  AzJO, by adding (curve to right) or subtracting (curve to left) 90° to/from AzB.
• Forward Computation from point J to point O using AzJO and R.

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

• Point G to point I to point O
• Point H to point J to point O
• Point H to point K to point O

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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