Relationship between azimuthal equidistant projection and WGS84 curvilinear to NED transformation

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What's the relationship between Azimuthal equidistant projection and WGS84 Curvilinear to ECEF to NED conversion?

In both transformations, I am assuming the reference point or the origin of the local tangent plane to be the first GPS measurement. Upon working on the transformation by hand, my results from Curvilinear to ECEF to NED does not match with results from azimuthal projection. Shouldn't curvilinear to ECEF to NED preserve distance given they are all Euclidean transformations?

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