# Creating CRS with coordinate origin at north-west corner of Mercator projection?

I want to define a custom CRS for a very special, technical use-case. For that I use a WGS84-World-Mercator as base definition, here are the `proj4` parameters taken von QGIS:

``````+proj=merc +lon_0=0 +k=1 +x_0=0 +y_0=0 +datum=WGS84 +units=m +no_defs
``````

For my custom CRS, I wish to have positive coordinates for any point in the world and all coordinates should be within the range of 0 and 1. The coordinate origin should be in the north-west corner of the Mercator.

To scale all coordinates within the range `[0, 1]`, I use the parameter `+to_meter=40075016.68557849` while leaving the scale factor `k_0=1` and removing the metric `units=m`.

``````+proj=merc +lon_0=0 +k_0=1 +to_meter=40075016.68557849 +x_0=0 +y_0=0 +datum=WGS84 +no_defs
``````

So far, so good, now the tricky part: How to create a coordinate origin at the north-west corner (North Pole)? Taking pen and paper, I found the solution is setting `+lat_0=90` and `lon_0=-180`, like that:

``````+proj=merc +lon_0=-180 +lat_0=90 +k_0=1 +to_meter=40075016.68557849 +x_0=0 +y_0=0 +datum=WGS84 +no_defs
``````

However, that creates weird results:

1. The `+lat_0=90` basicly does nothing to whatever value it gets set. Why?
2. The `+lon_0=-180` does behave like setting it to plus +180°, but I explicitly want the negative value from a mathematical view.

I know that plus and minus 180° are the same meridian on the map, but technically the difference with the minus is important for me.

How to create a CRS with coordinate origin at the north-west corner of the Mercator projection and only positive coordinates throught out the whole map? Is inverting the y-axis the issue here? Is chosing a World Mercator as base the wrong approach?

• I'm sure I'm misunderstanding, but you know that Mercator can't map the North Pole, right? It's coordinate would be infinite in the y direction. You could try doing a rigid rotation first, or, as you mention, using a different projection (eg, equiangular). – user1462 Oct 7 '16 at 14:24

``````+proj=merc +k_0=1 +to_meter=40075016.68557849 +x_0=20037508.34 +y_0=20037508.34 +datum=WGS84 +no_defs
• `+proj=merc +k_0=1 +to_meter=40075016.68557849 +x_0=20037508.34 +y_0=-20037508.34 +datum=WGS84 +axis=esu +no_defs` might be what you are looking for, but QGIS displays it upside down, – AndreJ Oct 8 '16 at 5:27