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I have a shapefile for world countries and I want to reproject it from GCS_WGS_1984 to Sphere_Eckert IV. However, it asks for Geographic Transformation; it says this is optional but it is required in order to proceed. I saw that the datum is D_Sphere but I cannot find a combination between this and WGS in the provided list.

Can please someone tell me what the right choice is? I use ArcGIS 10.1.

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ArcGIS (and really any Esri product) has no predefined datum transformations to convert to or from a sphere. The only thing that can really happen is to do an ellipsoid-to-ellipsoid conversion. There is an additional option to also change the geodetic latitudes to spherical latitudes which I'll describe below. There are several options to work around the Project Tool's requirement for a transformation.

  1. Define a custom transformation with the geoprocessing Create Custom Geographic Transformation tool. Pick the Null transformation if you don't want to do an ellipsoid-to-ellipsoid conversion. Otherwise, choose Geocentric Translations and leave the parameter values set to zeroes.

  2. Project the data in ArcMap. Either don't set a transformation at all (ignore the warning message if you get it) and save out a new feature class by right clicking the layer in the table of contents and selecting data, export data (use the data frame's coordinate system). You can also define a custom transformation in the Transformations dialog of the data frame's Coordinate System tab.

  3. If using a projection that has an Auxiliary Sphere version, use that (only Esri software will understand it though). Eckert IV has an Auxiliary Sphere version. These projections have an Auxiliary_Sphere_Type parameter. 0 - use semimajor axis of GeoCRS as sphere radius, 1 - use semiminor axis as radius, 2 - calculate and use authalic (equal area) radius, 3 - calculate and use authalic radius and convert geodetic latitudes to authalic latitudes.

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