Chapter B. Intersections

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.  

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:

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.