Finding the area of a region defined by a set of GPS coordinates, which represent the vertices of a polygon, typically involves converting those coordinates to a planar (flat) system and then applying a geometric formula like the Shoelace Formula. The core calculation method, as described in the reference, relies on a specific sequence of multiplications and summations based on the coordinate pairs.
Understanding the Coordinates
GPS coordinates are usually given as Latitude (Lat) and Longitude (Lon). These are spherical coordinates that define positions on the curved surface of the Earth. For accurate area calculations using planar geometric formulas, it's necessary to project these spherical coordinates onto a flat surface using a chosen map projection (like UTM, State Plane, etc.). Once projected, the coordinates are represented as Cartesian (X, Y) pairs, where X typically corresponds to Easting and Y to Northing. The method described below uses these planar (X, Y) coordinates.
The Calculation Method (Shoelace Formula)
The technique to calculate the area of a simple polygon given the planar (X, Y) coordinates of its vertices in order is known as the Shoelace Formula or Surveyor's Formula. The provided reference describes one specific variation of this formula:
To find the area of a polygon with ordered vertices $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$:
- List the coordinates of the vertices in order around the polygon, either clockwise or counterclockwise. Ensure the last vertex is connected back to the first to close the polygon. For the formula iteration, we consider the vertices indexed such that vertex i-1 is previous to vertex i, and vertex i+1 is next to vertex i. For the first vertex (i=1), the previous vertex is the last vertex (n), and for the last vertex (i=n), the next vertex is the first vertex (1).
- For each vertex, take its x co-ordinate and multiply it by the difference between the y co-ordinate of the next adjacent vertex and the y co-ordinate of the previous adjacent vertex. As the reference states, you "take each x co-ordinate and multiply it by the difference of the adjacent points' y co-ordinates, making sure you are consistent with the order of the subtraction and ensuring that you include the sign of the difference." This means for vertex $i$, you calculate $xi \times (y{i+1} - y_{i-1})$.
- Add together these products calculated in the previous step for all vertices.
- Divide this sum by two.
- Take the absolute value of the result. This absolute value is the area of the polygon.
Mathematically, this variation of the formula can be written as:
Area $= \frac{1}{2} \left| \sum_{i=1}^{n} xi (y{i+1} - y_{i-1}) \right|$
Where:
- $n$ is the number of vertices.
- $(x_i, y_i)$ are the coordinates of the $i$-th vertex.
- $(x{i+1}, y{i+1})$ are the coordinates of the next vertex (with $(x{n+1}, y{n+1}) = (x_1, y_1)$).
- $(x{i-1}, y{i-1})$ are the coordinates of the previous vertex (with $(x_0, y_0) = (x_n, y_n)$).
Step-by-Step Example
Let's calculate the area of a polygon with vertices at projected coordinates: (1, 1), (3, 4), (5, 2), and (2, 0).
Vertices:
- $P_1$: (1, 1)
- $P_2$: (3, 4)
- $P_3$: (5, 2)
- $P_4$: (2, 0)
*Applying the formula ($x_i (y{i+1} - y{i-1})$):**
| Vertex (i) | $x_i$ | $y_{i-1}$ | $y_{i+1}$ | $(y{i+1} - y{i-1})$ | Product $xi * (y{i+1} - y_{i-1})$ |
|---|---|---|---|---|---|
| 1 | 1 | $y_4=0$ | $y_2=4$ | $(4 - 0) = 4$ | $1 \times 4 = 4$ |
| 2 | 3 | $y_1=1$ | $y_3=2$ | $(2 - 1) = 1$ | $3 \times 1 = 3$ |
| 3 | 5 | $y_2=4$ | $y_4=0$ | $(0 - 4) = -4$ | $5 \times -4 = -20$ |
| 4 | 2 | $y_3=2$ | $y_1=1$ | $(1 - 2) = -1$ | $2 \times -1 = -2$ |
Sum of Products:
$4 + 3 + (-20) + (-2) = 7 - 22 = -15$
Final Calculation:
- Divide by two: $-15 / 2 = -7.5$
- Take the absolute value: $|-7.5| = 7.5$
The area of the polygon is 7.5 square units.
Important Considerations
While the Shoelace Formula provides an exact area for a polygon on a flat plane, using GPS coordinates introduces complexities:
- Earth Curvature: For large areas, treating the Earth as flat will introduce significant error. Survey-grade or GIS applications often use more sophisticated methods that account for the Earth's curvature and the specifics of the chosen map projection.
- Map Projection Distortion: Every map projection distorts area to some degree. The choice of projection impacts the accuracy of the calculated area, especially over large regions.
- Coordinate Accuracy: The accuracy of the final area calculation is directly dependent on the accuracy of the initial GPS measurements.
For precise cadastral surveying or large-scale mapping, specialized software and geodetic calculations are typically employed. However, for many practical purposes involving relatively small areas, converting Lat/Lon to a suitable projected coordinate system and using the Shoelace Formula is a common and effective approach.
Practical Applications
This method is fundamental in various fields:
- Surveying: Calculating land parcel areas.
- Geographic Information Systems (GIS): Determining the area of features like parks, lakes, or administrative boundaries.
- Mapping: Creating scaled maps and understanding spatial relationships.
By converting GPS coordinates to a planar system and applying the Shoelace Formula as described, you can effectively determine the area of a polygon defined by those coordinates.
