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I am working on the elevation data from Mapzen Global Terrain. When I save them into GeoTiff, there is a CRS option in which I use ProjectCRS: EPSG 4326 - WGS84. However, when looking into the Extent layer, the maximum value I get is 85.051128780 and minimum value I get is -85.051128780 instead of 90 and -90, respectively for Latitude range.

If I replace these 2 values in Extent layer, will the map extent to my desired range (-90 to 90 Lat and -180 to 180 Lon) with (-85.051128780 to -90 Lat and -180 to 180 Lon) and (85.051128780 to 90 Lat and -180 to 180 Lon) as Empty value or do they sketch the entire map to my desired range?

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No. Mapzen Global Terrain tiles will be designed for use in Web Mercator projection, which only goes to ±85.05.. degrees of latitude. From Wikipedia:

Because the Mercator projects the poles at infinity, a map using the Web Mercator projection cannot show the poles. Services such as Google Maps cut off coverage at 85.051129° north and south. This is not a limitation for street maps, which is the primary purpose for such services. The value 85.051129° is the latitude at which the full projected map becomes a square...

So it has one useful property: all tiles are square, including the one at zoom 0. But, the point at ±90° would be stretched to infinity in order to be rendered as a line along the top or bottom of the map. So the projection just cuts off before that point. That is why Antarctica looks so large, when it's really about the same size as Australia.

If you need estimates of elevation at the poles, you will need to look elsewhere. However if you only ever intend to do stuff on webmaps projected in Web Mercator, there's no point obtaining the remaining data since you won't be able to see it.

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