27.3 : Area Computation by the Alternative Coordinate Method

The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.

In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially around the traverse, ensuring the first corner’s coordinate is repeated at the end of the sequence to close the shape.

Solid diagonal lines are drawn from each x-coordinate to the subsequent y-coordinate, while dashed diagonal lines connect each y-coordinate to the subsequent x-coordinate. The products of these coordinates are calculated for both the solid and dashed lines:

• Solid-line contributions: Multiply each x-coordinate by the y-coordinate of the next point in the sequence.
• Dashed-line contributions: Multiply each y-coordinate by the x-coordinate of the subsequent vertex.

The area of the traverse is obtained by subtracting the total of dashed-line products from the total of solid-line products and dividing the absolute difference by two:

The alternative coordinate method is preferred in surveying for its simplicity, adaptability, and compatibility with manual and computer-assisted computations. Its systematic structure minimizes computational errors and offers a reliable way to handle complex traverses in geospatial analysis.