The standard 2-D rectangular coordinate system has an origin formed by the perpendicular intersection of the abscissa and the ordinate. Two systems, called the Math/Science Reference System and the Engineering/Surveying Reference System, are quite similar, but, from one system to the other, the reference axes are different and the direction of positive rotation is reversed (see Figure 4.2a and Figure 4.2b).
Math/Science Reference System
The 2-D coordinate system commonly used by mathematicians and scientists labels the abscissa as the X-axis and the ordinate as the Y-axis. The positive X-axis is considered to be the reference for angles, and rotation is counted positive counterclockwise. In this system, the Δχ and Δγ components of any directed line segment (vector) are
where
FIGURE 4.2 Math/Science and Engineering/Surveying Coordinate Systems
Engineering/Surveying Reference System
Surveyors, engineers, and others who work with mapping data often use a 2-D rectangular coordinate system, which is similar to the math/science system except that cardinal directions of north-south and east-west are superimposed upon the two axes and rotation is counted clockwise from north. In this system, the Ax (easting) and Ay (northing) components of any directed line segment are
where
The specific relationship between the math/science system and the engineering/ surveying system is that, in all cases, a + θ = 90°, which also means a = 90° – θ or θ = 90° – a. Being aware of these similarities and differences helps spatial data users make greater use of the polar/rectangular conversions hardwired into many calculators.
For example, most scientific calculators are hardwired according to the math/science system. To compute rectangular components of a line 100.00 meters long having an azimuth of 30°, the calculator will show 50.000 meters as being the north-south component when it is really the east-west component. Similarly, because it is using the math/science convention, the calculator will show 86.602 meters as being the east-west component when it really is the north-south component (in the engineering/surveying system). The only thing the user needs to do is switch the label of the computed components. A word of caution: each user should practice and work with known quantities to make sure the procedure being used is giving correct answers in the intended system.
In each case, the 2-D math/science reference system and the 2-D engineering/ surveying reference system can be expanded into a 3-D system by adding a Z-axis. The convention adopted in this topic is to use e/n/u for the local reference frame because it is right-handed and Äe/Än/Äu can be conveniently rotated into the ECEF right-handed X/Y/Z reference frame. Regretfully, azimuth in the engineering/surveying system is not consistent with the right-hand rule.
