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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

 

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