Calculating Projection Distortion

BrianZ111BrianZ111 Global Mapper UserPosts: 40Trusted User
edited December 2014 in Projection Questions
I created a spreadsheet where I paste in the coordinates of vertices in a feature and the great circle measurements of that feature's segments. For distance distortion the spreadsheet finds the distance of the segments in the projection using the coordinates and subtracts the measured great circle distances. For angular distortion it finds the angle between two segments in the projection and subtracts the angle between the great circle paths. The angle between the great circle paths is determined by subtracting the initial bearings measured for the segments from the central point.

So far so good I think?

Now I'd like to know what the maximum distortion in angular distance is in a given area for a projection. Using trigonometry, I take the angular distortion difference above as a central angle, and the length of one segment as the two adjacent sides. The calculated 3rd side length is the angular distance at the end of the segment.

Will using the 90° corners of the perimeter segments give me the maximum angular distance due to distortion or do I need to measure a different angle? Is this something that even makes sense to calculate? I'm just trying to get a feel for the maximum error caused by different projections in different size areas in a quantifiable way. Is there an easier way to calculate the distortion of a projection over a given area, either within global mapper or a 3rd party program or spreadsheet?

Here's the spreadsheet I made: http://www.zagerdesign.com/temp/projection_distortion_calculation.xlsx. It has coordinates and measurements pasted in on the Map Perimeter Measurements sheet and the calculations are on the Projection Distortion sheet.

Thanks for any help pointing me in the right (undistorted :D) direction.

Comments

  • GeoGeo Global Mapper User Posts: 92Trusted User
    edited December 2014
    hello BrianZ111,

    Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

    I hope that it can help you.
  • BrianZ111BrianZ111 Global Mapper User Posts: 40Trusted User
    edited December 2014
    Yes I was aware of Tissot as a visual. I guess I'm looking for the math behind it but I didn't think to google that exactly...so thanks. If you know of a good page that explains the math let me know, otherwise I'll google around and figure it out.
  • GeoGeo Global Mapper User Posts: 92Trusted User
    edited December 2014
    There's a one-to-one correspondence between the Tissot indicatrix and the metric tensor of the map projection coordinate conversion.[1]

    References
    1. Goldberg, David M.; Gott III, J. Richard (2007). "Flexion and Skewness in Map Projections of the Earth". Cartographica 42 (4): 297–318. doi:10.3138/carto.42.4.297. Retrieved 2011-11-14.
  • BrianZ111BrianZ111 Global Mapper User Posts: 40Trusted User
    edited December 2014
    OK, thanks. I will investigate that.
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