7.2 Area Computations (DMD, DPD, Coordinates) and Subdivision/Consolidation Survey Standards

7.2 Area Computations (DMD, DPD, Coordinates) and Subdivision/Consolidation Survey Standards

Accurate area calculation of boundary lots is a primary legal responsibility of Geodetic Engineers. Under Philippine property law and land registration statutes (PD 1529), land titles specify lot areas to the nearest square meter. This section covers classical numerical methods (DMD and DPD), modern coordinate geometry methods, treatment of circular curved boundaries, and DENR administrative regulations for lot subdivision and consolidation surveys.

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Double Meridian Distance (DMD) Method

The Double Meridian Distance (DMD) of a course is defined as the sum of the meridian distances of its two endpoints. The meridian distance of a point is the perpendicular distance from that point to the reference north-south meridian passing through the most westerly station of the traverse.

Rules for Computing DMDs:

1. Rule 1 (First Course): The DMD of the first course equals the departure of the first course itself. $$\text{DMD}_1 = Dep_1$$
2. Rule 2 (Succeeding Courses): The DMD of any succeeding course equals the DMD of the preceding course, plus the departure of the preceding course, plus the departure of the course itself. $$\text{DMD}_i = \text{DMD}_{i-1} + Dep_{i-1} + Dep_i$$
3. Rule 3 (Check / Last Course): The DMD of the final course must be equal in magnitude to the departure of the final course, but opposite in sign. $$\text{DMD}_n = - Dep_n$$

Area Formula using DMD:

$$\text{Double Area } (2A) = \sum_{i=1}^{n} (\text{DMD}_i \cdot L_{adj,i})$$ $$\text{Area } A = \frac{1}{2} \left| \sum_{i=1}^{n} (\text{DMD}_i \cdot L_{adj,i}) \right|$$

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Double Parallel Distance (DPD) Method

The Double Parallel Distance (DPD) method is the orthogonal counterpart to DMD, using an East-West reference parallel (passing through the most southerly station) and working with latitudes instead of departures.

Rules for Computing DPDs:

1. First Course: $\text{DPD}_1 = L_1$
2. Succeeding Courses: $\text{DPD}_i = \text{DPD}_{i-1} + L_{i-1} + L_i$
3. Check (Last Course): $\text{DPD}_n = - L_n$

Area Formula using DPD:

$$\text{Double Area } (2A) = \sum_{i=1}^{n} (\text{DPD}_i \cdot Dep_{adj,i})$$ $$\text{Area } A = \frac{1}{2} \left| \sum_{i=1}^{n} (\text{DPD}_i \cdot Dep_{adj,i}) \right|$$

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Coordinate Method (Gauss's Shoelace Formula)

With computer automation and total station/GNSS field data, the Coordinate Method is the standard approach for area determination. Given a closed polygon with $n$ vertices having ordered grid coordinates $(E_1, N_1), (E_2, N_2), \dots, (E_n, N_n)$ listed sequentially in either clockwise or counter-clockwise order around the perimeter:

$$2A = |(E_1 N_2 + E_2 N_3 + \dots + E_n N_1) - (N_1 E_2 + N_2 E_3 + \dots + N_n E_1)|$$

In matrix / shoelace expansion form:

$$2A = \left| \sum_{i=1}^{n} (E_i N_{i+1} - E_{i+1} N_i) \right| \quad \text{where } (E_{n+1}, N_{n+1}) = (E_1, N_1)$$

Alternatively, using coordinate differences:

$$2A = \sum_{i=1}^{n} E_i (N_{i+1} - N_{i-1}) \quad \text{or} \quad 2A = \sum_{i=1}^{n} N_i (E_{i+1} - E_{i-1})$$

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Area of Irregular and Curved Boundaries

Lots bounded by circular curves (such as road right-of-ways, cul-de-sacs, or river arcs) require combining polygon traverse area with circular segment corrections.

1. Circular Segment Correction

For a lot boundary following a circular curve with radius $R$ and central angle $I$ (or $\Delta$):

• Central Angle in Radians: $I_{rad} = I^\circ \times \frac{\pi}{180^\circ}$
• Area of Circular Sector: $$A_{\text{sector}} = \frac{1}{2} R^2 I_{rad} = \frac{\pi R^2 I^\circ}{360^\circ}$$
• Area of Sector Triangle: $$A_{\text{triangle}} = \frac{1}{2} R^2 \sin(I)$$
• Area of Circular Segment: $$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{1}{2} R^2 \left( I_{rad} - \sin(I) \right)$$

Rule of Application:

• If the circular arc lies outside the straight chord joining the two lot corners (convex lot curve), ADD the segment area $A_{\text{segment}}$ to the polygon area calculated via DMD or Coordinates.
• If the circular arc curves inward toward the lot interior (concave lot curve, like a setback curve), SUBTRACT the segment area $A_{\text{segment}}$ from the polygon area.

2. Irregular Water Boundaries (Trapezoidal & Simpson's 1/3 Rules)

When a boundary follows a natural shoreline or river bank, perpendicular offsets $h_0, h_1, h_2, \dots, h_n$ are measured at uniform intervals $d$ along a base line:

• Trapezoidal Rule: $$A = d \left( \frac{h_0 + h_n}{2} + h_1 + h_2 + \dots + h_{n-1} \right)$$
• Simpson's 1/3 Rule (requires an even number of intervals $n$, i.e., odd number of offsets): $$A = \frac{d}{3} \left[ (h_0 + h_n) + 4(h_1 + h_3 + \dots + h_{n-1}) + 2(h_2 + h_4 + \dots + h_{n-2}) \right]$$

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DENR Standards for Subdivision and Consolidation Surveys (DAO 2007-29)

Under the Philippine Department of Environment and Natural Resources (DENR) Memorandum Circulars and DAO 2007-29 (Revised Regulations on Land Surveys):

1. Classification of Land Surveys

2. Corner Markers & Monumentation Rules

All lot boundary corners established or re-established during cadastral or property surveys must be marked on the ground with standard monuments:

• Standard Corner Monument: Private Survey (P.S.) cylindrical concrete monument.
• Dimensions: 15 cm in diameter by 50 cm in length.
• Installation: Set vertically in the ground with 45 cm buried below ground surface and 5 cm projecting above ground level.
• Material Composition: Cast concrete with a volumetric mix ratio of 1 part Portland cement, 2 parts clean sand, and 4 parts gravel (1:2:4 mix).
• Center Mark: The exact corner point is marked on top of the monument by a galvanized iron (G.I.) nail/spike centered in a shallow cross mark.
• Bureau of Lands Monuments (B.L.): Primary control corners use $15 \times 15 \times 60\text{ cm}$ square or $20\text{ cm}$ diameter cylindrical monuments.

3. Tie-Line Requirements

Every isolated lot or subdivision survey plan submitted to the Land Management Bureau (LMB) or Land Registration Authority (LRA) must be mathematically connected by a Tie-Line (bearing and distance) from Corner 1 of the lot to an official reference monument, such as a Bureau of Lands Location Monument (BLLM), City Boundary Monument (CBM), or established triangulation station.

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Worked Example: DMD Area Computation

Using the adjusted traverse data from Section 7.1:

Step 1: Compute DMD for each Course

• $\text{DMD}_{1-2} = Dep_{1-2} = \mathbf{+50.024\text{ m}}$
• $\text{DMD}_{2-3} = \text{DMD}_{1-2} + Dep_{1-2} + Dep_{2-3} = 50.024 + 50.024 + 129.940 = \mathbf{+229.988\text{ m}}$
• $\text{DMD}_{3-4} = \text{DMD}_{2-3} + Dep_{2-3} + Dep_{3-4} = 229.988 + 129.940 + (-84.824) = \mathbf{+275.104\text{ m}}$
• $\text{DMD}_{4-1} = \text{DMD}_{3-4} + Dep_{3-4} + Dep_{4-1} = 275.104 + (-84.824) + (-95.140) = \mathbf{+95.140\text{ m}}$ (Check: $\text{DMD}_{4-1} = +95.140$, which equals $-Dep_{4-1} = -(-95.140) = +95.140\text{ m}$. Check passes!)

Step 2: Compute Double Areas ($2A = \text{DMD} \times L_{adj}$)

Step 3: Compute Final Lot Area

$$\text{Double Area } 2A = |-29,298.055\text{ m}^2| = 29,298.055\text{ m}^2$$ $$\text{Area } A = \frac{29,298.055}{2} = \mathbf{14,649.03\text{ m}^2} \quad \text{or} \quad \mathbf{1.4649\text{ hectares}}$$