Planar vs. Euclidean distance measurements

[Craig_Eissler_Iceman]

I understand Planar vs. Geodesic measurements and Euclidean vs. Geodesic measurements.

However, is there a difference between Planar and Euclidean measurements, or are these just interchangeable terms that ArcGIS uses?

[AyanPalit · Accepted solution]

@Craig_Eissler_Iceman my notes on this topic:

Euclidean distance is the straight-line distance between two points in Euclidean space (in theory this can be 2D or 3D). The distance is calculated using the Pythagorean theorem, relating the difference in coordinates along each axis.

Reference: Understanding Euclidean distance analysis

Planar distance is straight-line Euclidean distance calculated in a 2D Cartesian coordinate system. Here calculations are always made on a flat, 2D plane; thereby planar distance equals Euclidean distance in 2D Euclidean space but may differ if  Euclidean space is 3D.

Reference: Geodesic versus planar distance

[Craig_Eissler_Iceman]

Thanks, yes, I had already read all of that.

So I guess I should have re-framed my question to say, "why" is there potentially a difference between the two in 3D space?

[LetaFranklin]

Planar distance is like measuring the path along the surface accounting for the Earth's curvature but ignoring elevation changes. It treats the earth if it were a perfect 2D plane.

Euclidean distance is the direct, straight-line measurement, cutting through the Earth as if tunneling between the two points.

If you were flying between San Francisco and New York, your flight path would follow a curved trajectory due to the Earth’s shape (similar to planar distance). This is how a crow would fly between the two cities, if it could fly without stopping for breaks, of course. A Euclidean distance is a straight line through the earth, ignoring the planet’s curvature entirely. AI tells me that tunnel would be 6-9 miles below the surface at the midway point 🙂 

[Craig_Eissler_Iceman]

Hmmm, OK, well if you're correct, that would explain the difference between Planar and Euclidean.

But you say, "Planar accounts for the curvature of the Earth". If so, then what's the difference between that and Geodesic? ...I ask because I've always thought it was Geodesic that accounts for the Earth's curvature? 

Or, are you saying that this is just the difference in 3D, not 2D?

[JoshuaBixby]

Focus on @AyanPalit's response, other responses offered today have either been wrong or poorly worded enough to be misleading.

[Craig_Eissler_Iceman]

...Yes, but my follow-up question to that was, "What is it that makes the possible difference in 3D"?

Admittedly, I am not familiar with using 3D in GIS.

Is the answer "elevation"? If so, that might make some sense.

[AyanPalit · Accepted solution]

@Craig_Eissler_Iceman 

The Euclidean space can be 2D or 3D. Considering the 3D Cartesian coordinate system below, planar distance will be on the 2D orange plane. Note that, Euclidean distance can be calculated using the Pythagorean theorem in the 2D orange plane.

Now consider 2 points on the green plane that are at different y-coordinates, and project them back to the orange plane (flattening them in the process). In this situation, the planar distance calculated on the projected orange plane will be different than the Euclidean distance calculated on the green plane. This example illustrates planar distance equals Euclidean distance in 2D Euclidean space, but may differ if Euclidean space is 3D.

With advances in data collection technologies, majority of GPS/GIS/location information, includes z-coordinates representing the elevation. ArcGIS has various capabilities around storing, managing, displaying as well as analysis like measurements using 3D data. Here's a good read:  Why you should think about coordinate systems when working in 3D

[Craig_Eissler_Iceman]

OK Ayan, the illustration with description helps.

But two things:

1. Can you give me a real-life example? ...Or, can I just think of Z being elevation, to make things simple to visualize?

2. Why does it seem Euclidean is the term used for raster analysis and Planar for vector analysis? ...Or, is this merely coincidence?

[AyanPalit]

@Craig_Eissler_Iceman 

1. Euclidean space can be 2D or 3D. Elevation is a great example in real world and is used in GIS modelling, digital twins etc. In engineering / drafting, height is primarily along the Z-axis, defining the vertical dimension of 3D objects.
2.  The mathematical concepts of Euclidean geometry have wide and varied applications. How it's used in a specific software package like ArcGIS, AutoCAD and various modules therein, can vary. In short, Euclidean distance can be applied to any physical space (3D or 2D that is just a version of 3D, not using the third dimension); the concept can be applied to vectors and rasters.