# How can I calculate degree of distortion on an Azimuthal Equidistant projection?

The Tissot's indicatrix of an Azimuthal Equidistant projection shows ellipses being stretched along latitudes.

How can I calculate the degree (amount) of this distortion? That is, what is the ratio of major axis to minor axis of a Tissot ellipse as a function of latitude (assuming the center of projection is the north pole)?

Edit: Found numerical values on the GIS help page.

``````Degrees from center     Percentage of scale distortion along parallels
15                      1.2
30                      4.7
45                      11.1
60                      20.9
90                      57
``````

I can interpolate this well and get what I need. But would still be good to know what the analytical formula is. Feels like a function of `cos(latitude)` since an Azimuthal Equidistant projects latitudes with linearly increasing distance from center instead of cosine.

You can find the scale distortion from the latitude with this simple formula (latitude in radians) given an origin at the North Pole:

``````(pi/2) - lat
------------
cos(lat)
``````

This will give you the ellipse's ratio you're looking for.

Example with 60°N:

``````(pi/2) - (pi*60/180)
--------------------   = 1.04719...
cos(pi*60/180)
``````

Which is a 4.7% distortion as stated in ArcGIS Help.