Cálculo de áreas por el método de las coordenadas - Topografía

Main ideas / concepts

• The video explains the coordinate method for calculating the area of topographic figures (polygons) using the northing (North) and easting (East) coordinates of the polygon’s vertices.
• The coordinate method is presented as more precise than other area-calculation methods when coordinates are available.
• Two equivalent formulations are covered:
  1. “Mechanized” (Gauss/Hitler-style) method
  2. Matrix method
• In both methods, the computations produce “double the area”; the final area is obtained by dividing by 2.
• If the computed value is negative, the area is taken as the absolute value (area itself cannot be negative).

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Method 1: Mechanized coordinate method (Gauss/“Hitler” mentioned)

Data organization

• Start from an ordered table of polygon vertices with columns:
  • point, north, east
• Arrange the data so that:
  • You include extra rows/entries to wrap around the polygon (the “next” vertex of the last one returns to the first).
  • In the described example, additional rows are generated so each north can be paired consistently with the corresponding upper and lower east values to close the polygon loop.

Core computation rule (double area)

• For each vertex (north coordinate), multiply it by:
  • the easting immediately above the vertex in the table, and
  • the easting immediately below the vertex in the table.
• Do not multiply by the “frontal” easting (as emphasized).
• For each vertex’s contribution, compute a difference of products. The method effectively uses:

[ (N_i \cdot E_{\text{upper}}) - (N_i \cdot E_{\text{lower}}) ]

• Then sum these contributions algebraically across all vertices (so signs matter).

Handling signs and final area

• The sign of the intermediate algebraic sum depends on the subtraction order.
• If the total computed value (which equals double the area) is negative, take its absolute value.
• Then compute the final area:

[ A = \frac{|2A|}{2} ]

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Method 2: Matrix method

Relationship to the first method

• This method is described as very similar to the mechanized method.
• The main difference is how the products are organized and presented (arranged “diagonally like a matrix”).

Data organization and diagonal products

• Use the ordered north/east table for the polygon vertices.
• Compute products in a diagonal pattern (matrix-like):
  • Products to the right use positive signs
  • Products to the left use negative signs
• Sum the products algebraically.

Output meaning

• As with the mechanized method, the algebraic sum equals double the area.
• The final area is obtained the same way:
  • take the absolute value if negative, then divide by 2.

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Example workflow applied to a polygon (outline)

• A polygon is shown on a Cartesian topographic plan with vertices labeled a, b, c, d, ….
  • The worked example later treats it as a 5-vertex traverse: 1 through 5.
• The polygon’s vertex coordinates (northing and easting) are given in a table.
• Steps to apply the coordinate method:
  1. Choose the coordinate method since coordinates are available and it is stated to be preferable/more precise.
  2. Recreate/reformat the coordinate table with headings: point | north | east.
  3. Create extra rows to “close” the polygon loop (so eastings for the wrap-around vertex are included).
  4. Using the mechanized approach, multiply each north by:
     • the easting above and
     • the easting below (difference-based).
  5. Follow an emphasized practical recommendation:
     • compute operations inside parentheses first,
     • then subtract as indicated,
     • then multiply by the north values,
     • then take the algebraic sum.
  6. The resulting value corresponds to double the area.
  7. If negative, take the absolute value.
  8. Divide by 2 to get the final polygon area.

Worked result

• Final computed polygon area (as stated): 95,627 m²

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Speakers / sources featured

• No specific speaker name is identified in the subtitles (the narrator/teacher is not explicitly named).
• Referenced methods/sources:
  • Gaudí (mechanized method)
  • Hitler (mechanized method)
  • Matrix / Matrix method concept (as related to geometry courses on matrices)