Shoelace formula

Using the Gaussian trapezoidal formula (after Carl Friedrich Gauss ), it is possible to calculate the area between several line related to a measurement / coordinated points, so for example the area of ​​a simple polygon. By splitting this area into individual related to the measurement line Trapeze start the calculation.

Word formula: Double the surface corresponds to the sum of the current right value and the subsequent, multiplied by the difference between the current high value and high value following.

Or:

The dual surface is the sum of the current high value and the subsequent, multiplied by the difference of the following legal value and current legal value.

Where the indices that are greater than ever modulo must be considered, ie with mean.

If the points are traversed in the direction of rotation of the coordinate system, the calculated area is positive, otherwise negative.

Example

The area of ​​the right-wing image is to be calculated using the trapezoidal rule. It is a geodetic coordinate system is used in the positive direction of rotation corresponds to the clockwise direction. To obtain a positive surface area, therefore, the points must be traversed clockwise.

The documentation of the calculation course can be done in different ways. To simplify this notation are used, for example, within the surveying prefabricated forms.

If we decompose the image to be seen on the single surface in the four sub-areas and, we obtain the following formula

Is therefore

Due to small deformations one obtains

This result corresponds to the above trapezoidal formula.

Triangle formula

The Gaussian triangle formula is obtained by factoring out and rearranging the trapezoidal formula. The indices that are greater than must be considered modulo again, ie with and is meant to mean.

Analogy can be

To

Reshape.

In words, this formula reads:

The dual surface is the product of the current legal value and the difference from previous high- value and high- value follows.

Or:

The dual surface is the product of the current high value and the difference from previous Easting and following Easting.

Application

For area estimation in the Gauss -Krüger coordinate system, the surface distortion must be taken into account by the distance to the main meridian dependent.

Formula to take account of surface distortion:

  • = Area reduction
  • = Initial surface
  • = Distance to the main meridian
  • = Radius of the earth ( geodetic Earth radius = 6381 km )
  • Plane Geometry
  • Mathematical Geography
  • Carl Friedrich Gauss
Easting and northing
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