### C. Three-tangent arc

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

Figure I-5 |

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

Figure I-6 |

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

Figure I-7 |

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, d_{GH}.

Using the azimuths, compute deflection angles Δ_{G} and Δ_{H} 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 d_{GJ} and d_{GI} are the tangent distances for the first curve; distances d_{HK} and d_{HJ} 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 Az
_{B}and tangent distance d_{GJ} - Compute azimuth to the radius point, Az
_{JO}, by adding (curve to right) or subtracting (curve to left) 90° to/from Az_{B}. - Forward Computation from point J to point O using Az
_{JO}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