### 1. Solving point positions

#### a. Connecting an unknown position

A point whose absolute position is to be determined has two unknowns: North and East. To solve both unknowns requires two measurements connecting the point to a set of known coordinates. In the case of a point-to-point forward computation, the unknown point is connected with a length and direction to a single known point, Equations A-1 and A-2.

The two measurements don't have to be from a single point. In Figure B-1, the coordinates of points L and M are known, point P's are not.

Figure B-8 |

The forward computation Equations A-1 and A-2 can be written for point P from points L and M:

Equation B-1 | |

Equation B-2 |

Equations B-2 and B-8 have four unknowns: L_{LP}, Dir_{LP}, L_{MP}, and Dir_{MP}.

#### b. Intersections

To solve the position of point P, it must be connected to points L and M using one of the following combinations:

- L
_{LP}and Dir_{MP} - Dir
_{LP}and L_{MP} - L
_{LP}and L_{MP} - Dir
_{LP}and Dir_{MP}

These are the standard COGO *intersections* since point P is at the intersection of two measurements. There are three intersections based on the type of measurements:

- Distance-direction (or Direction-distance, we'll use these interchangeably)
- Distance-distance
- Direction-direction

Fixing two of the measurements allows solution of the other two. For example, given the coordinates of points L and M, and the azimuths from both to point P:

Point | North (ft) | (East) | Azimuth to P |

L | 614.80 | 2255.90 | 117°22'40" |

M | 791.53 | 2517.03 | 198°10'30" |

Equations B-1 and B-2 can be written as:

Equation B-3 | |

Equation B-4 |

These can be solved simultaneously for L_{LP} and L_{MP}. Either distance can then be substituted back into its respective side of Equations B-3 and B-4 to perform a forward computation to point P.

The problem is that while a direction-direction intersection is relatively easy to solve simultaneously, the other two types are not. The two equations for a direction-distance intersection will contain the sine and cosine of one unknown direction; distance-distance intersection equations will contain the sine and cosine of two unknown directions. Because sine and cosine functions are non-linear, these are not trivial solutions.

So how do we go about solving direction-distance and distance-distance intersections?

#### d. Solution Methods

For manual solutions, there are two general ways to solve COGO intersections:

- Triangle-based
- Arc-based

For both, we'll look first at its underlying geometry then how it's applied for the different types of intersections.