Shouldn't the geodesic area always be greater than the planar area of a polygon?


Tested in ArcMap and ArcGIS Pro, but several my polygons in a shapefile are having projected areas that are greater than the geodesic area.

I've loaded in a shapefile that has a GCS of WGS1984; the dataframe is set to GCS: WGS1984. Once loaded I perform the following:

Geodesic Area

  • Add Field to attribute table and set to double
  • Right click field and selected Field Calculator
  • Supply the following Python input: !Shape!.getArea("GEODESIC", "SQUAREMETERS")

Projected Area

  • Select the Project tool in Data Management > Projections and Transformations
  • Project the layer to PCS: World Sinusoidal

Then I use Python to check the 2 fields against one another. A quick check of:

# pseudocode
for each polygon:
    diff = geodesic_value - projected_value

A negative value for diff means that projected was larger; postive is the inverse. It was my understanding that geodesic would (and should) always be larger.

  • I don't see why that should be the case, but if it should, then what you have here is a bug report submitted to the Internet, not Tech Support. – Vince Nov 26 '19 at 0:17
  • @Vince My need for accuracy here comes from a comment you made that the geodesic area is the correct way to calculate the area of a polygon on the surface of the Earth. As I attempt to replicate that I am looking at how ArcGIS handles different area calculations. If the geodesic area is not always great than the planar area, is it correct to say the geodesic area is the true area and the planar area is an accurate area? Meaning that it may be larger or smaller but not vastly different. – datta Nov 26 '19 at 0:24
  • You may have read too much into my comment. Given a global dataset, geodetic calculation is preferred over any global projection, but I'm not going to engage on the concept of true area. – Vince Nov 26 '19 at 0:31
  • @Vince Why is it preferred over global projection? – datta Nov 26 '19 at 14:32
  • Because you already have the data in a geographic CRS. – Vince Nov 26 '19 at 14:48

It was my understanding that geodesic would (and should) always be larger.

I don't agree.

A polygon feature is defined by its vertices.

The area enclosed by the sides following geodesic lines is not the same as the area enclosed by the sides following straight lines in world sinusoidal projection.
It may be one greater than the other, depending on the direction (outwards or inwards of the polygon) the curvature of each geodesic line between the vertices of the polygon results.

Densify your polygons. The difference should tend to zero when the distance between vertices tends to zero.

| improve this answer | |
  • So it is not an accurate statement to say that the geodesic area is always greater than the planar area? – datta Nov 26 '19 at 0:21
  • 1
    It is just a false statement. Longlat Polygon ((10 5, 5 25, 15 10, 10 5)) could be an example. – Gabriel De Luca Nov 26 '19 at 0:33

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