I have multiple datasets in my ArcGIS, each having different map projections. All of them are vectors. Some are using Sphere Mercator, others using WGS84. I cannot perform the reprojection function in ArcGIS as I do not have the formula for geographic transformation. I am stuck and cannot proceed to make a map due to this issue. The problem is almost certainly with the ‘Sphere_Mercator’ data. This projected coordinate reference system (CRS) is using a sphere-based geographic CRS. ArcGIS doesn’t have any predefined geographic/datum transformations that convert to or from a sphere. The data may have been built on the sphere, or it might have been built on WGS 1984 and then projected to the Sphere_Mercator CRS. If you have reference data in WGS 1984 that you can compare the Sphere_Mercator data to, add both to ArcMap. Set the data frame to use the WGS 1984-based CRS. Does the Sphere_Mercator line up correctly? If so, unproject it to ‘Sphere’ GeoCRS using the Project Tool, then use the Define Projection tool to change the CRS to WGS 1984. Then use the Project Tool to convert it to EPSG:3857. Source.