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:
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.
The alternative coordinate method computes the area within the traverse using its x and y coordinates, which are arranged sequentially around the shape.
Each corner's coordinates are written as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator, listed horizontally in order. The first corner's fraction is repeated at the end of the sequence.
Draw solid diagonal lines connecting each x-coordinate to the next y-coordinate in sequence. Similarly, draw dashed diagonal lines connecting each y-coordinate to the next x-coordinate in the same sequence.
Calculate the solid-line contributions by multiplying each x-coordinate with the next y-coordinate.
Similarly, multiply each y-coordinate by the next x-coordinate to compute the dashed-line contributions.
Twice the area within the traverse is calculated by subtracting the total of the dashed-line products from the total of the solid-line products.
Take the absolute value of this difference and divide it by two to determine the exact area within the traverse.
This method is widely used for its simplicity, accuracy, and adaptability for manual and computer-based calculations in surveying.
繁體中文
