### 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 |

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 Az_{QO} and Az_{QP}, 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 P
_{1}and P_{2}

Determine the direction of a line tangent to an arc.

Figure I-3 |

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 P_{1}, perform a Forward Computation from point Q:

Equation I-13 | |

Equation I-14 | |

Equation I-15 |

To compute the coordinates of point P_{2}, perform a Forward Computation from point Q:

Equation I-16 | |

Equation I-17 | |

Equation I-18 |