3. Area By DMDs

a. Concept

Software uses coordinates for area determination. Most surveyors who might solve area manually generally use the coordinate method also. A traditional method of computing area of a closed polygonal traverse is by DMDs: Double Meridian Distances. While rarely used anymore, every now and then you'll hear it mentioned so we'll spend some time going over Area by DMDs.

A meridian distance (MD) is the distance from a reference meridian to the center of a line; it is measured in the East (X) direction, Figure G-10.

img23

Figure G-10
Meridian Distances

 

The reference meridian used can be placed anywhere, but generally, it passes through the first point of the traverse. Triangular and trapezoidal areas can be computed by multiplying a meridian distance by the latitude of the respective line. The base of each triangle or trapezoid coincides with the meridian. Adding the areas, some of which are negative based on the latitude and meridian distance, results in the area of the polygon.

Why then the Double Meridian Distance? Well if we start at point A in Figure G-10 and begin computing meridian distances, we see a pattern start to develop, Equation (G-4).

img24

Eqn (G-4)

 

See all those 1/2's? By multiplying both sides of each MD equation by 2, the 1/2's all go away and we're left with Double Meridian Distances (DMDs) on each left side, Equation (G-5).

img26

Eqn (G-5)

 

Notice that the DMD of the last line is the same as the Departure of that line except with an opposite math sign. That's the math check.

Multiplying each line's DMD by its Latitude and summing the results will give us double the area.

img25

Eqn (G-6)

 

Just like area by coordinates it looks pretty confusing, but there is a pattern that helps guide the calculations.

You'll sometimes see reference to Area by DPDs (Double Parallel Distance). This is identical to the DMD method except everything is rotated 90°. Parallel distances are determined from an E/W reference line (a parallel) and multiplied by the latitudes.

b. Examples

(1) Traverse 1

Figure G-11 showns the example Bearing Traverse with the previously computed adjusted Latitudes and Departures.  

img15 
  Adjusted
Line Lat (ft) Dep (ft)
AB -176.386 -438.574
BC +203.382 -73.105
CD +192.340 +198.635
DE -219.336 +313.044
 
Figure G-11
Example DMD Traverse
 

 

Step (1) Compute the DMDs

Starting with line AB and use Equation (G-4) to computed the DMDs.

img28

If you examine the Deps and DMDs computations above, you should see a pattern to the computations. This pattern is shown using colored arrows below.

img29

 

Step (2) Multiply DMDs by Lats; add the products 

  Adjusted    
Line Lat (ft) Dep (ft) DMD (ft) DMD x Lat (sq ft)
AB -176.386 -438.574 -438.574 +77,358.3
BC +203.382 -73.105 -950.253 -193,264.4
CD +192.340 +198.635 -824.723 -158,627.2
DE -219.336 +313.044 -313.044 +68,661.8
     

sum:

-205,871.5

 

Step (3) Compute the area using Equation (G-6)

img39

Note that this is the same as computed by coordinates. Surprise.

 

(2) Traverse 2

What about the Crossing Traverse?

Figure G-12
Parcel With Obstructions

Figure G-13
Crossing Traverse For the Parcel

 

The parcel area, Figure G-12, can not be determined by DMDs without additional computations. By definition, Area by DMDs is limited to travel along the traverse path so you would be determining the area of E-F-G-H-E, Figure G-13. That area doesn't make sense since the traverse crosses itself. To compute the area of the parcel by DMDs, you would need to determine the Lat and Dep of lines EF and FH, then compute DMDs around the perimeter. By the time all that was done, the area could have been computed by coordinates.