Shift country polygons to other world-regions in QGIS, adapting it's size corresponding to the true area

Question

On the website The true size of ..., you can shift the shape of countries to other world regions to see the true size (area) on a distorted map (mercator projection). Dragging Greenland over Africa is a classic example (see screenshot below).

Unfortunately, on the site I can't find any information about the procedure in the background - how it's done.

I'm wondering how the same result can be achieved using QGIS only. What I tried and what works as a workaround: Save a screenshot (like the one below), georeference this image (based on the background basemap, using control points in Africa: Juba, Kampala, Nairobi, Harare, Lusaka etc.) and than, on the georeferenced image, vectorize the shape of Greenland.

However, I would like to find a solution using QGIS only, without importing the data. How could this be achieved?

Edit:

What I tried 2.0 / An idea is to use the following procedure:

1. Set project CRS to Web-Mercator (EPSG:3857)

2. Re-project the same data (country polygons) to a Web-Mercator (EPSG:3857) and an equal area CRS, e.g. EPSG:8857 to have a layer in each of these CRS. I used the pre-installed basic world countries polygon layer in EPSG:4326 (type world to the coordinates field at the very bottom of the QGIS main window).

3. Use the icon Move feature from Advanced Digitizing Toolbar and move a feature from the equal area layer (see screenshot). This returns a "true size" version (but heavily distorted in shape) of the original polygon:

Yellow: layer in equal area projection, blue in Mercator - when shifted, the polygon from the equal area layer mainaints the "true size", but the original distortion, inherent in the projection, as well:

To keep the shape, you could do the same with the other layer (in Web-Mercator). The resulting polygon would preserve the shape, but be extremely distorted in size (Greenland almost the same size as Africa). At this point, the new Feature scaling tool (available since QGIS 3.18) could be used. However, how to set an appropriate scale factor? So I guess there must be better options to achieve that.

Again the blue layer is in Mercator: it preserves shape when shifted, but size is extremely blowed up as it is inherent to Mercator projection:

Answer 1

A relative simple solution would involve these steps:

1. On the layer in Web Mercator projection (EPSG:3857), create a new attribute with field calculator, calclulating the area distortion of each polygon: divide the distorted area by the "real" area. The distorted area can simply be calculated with area($geometry) (planimetric calculation). The real area can either be taken from pre-existing data (or attributes) or as well be calculated in the field calculator using $area - this returns an ellipsoidal area (or could be calculated on another layer, re-projected to an equal area CRS). Thus to get the scale factor (area distortion), it in enough to create a new attribute with this expression: area($geometry) / $area on the layer in Web Mercator.

2. Now use Menu Processing / Toolbox / Affine Transformation an set the Scale factor for x- and y-axis each to sqrt( 1/ "scale_factor"). You want to shrink the areas, thus 1/scale factor. As shrinking will be done in x- and y-direction (two-dimensional) and both together should result in the scale-factor, the one-dimensional shrink-factor (for x/y-axis each) is the square root (sqrt) of the scale factor for the area.

Screenshot: for Greenland, the two-dimensional scale-factor calculated is 16.33. When applying the affine transformation with the formula from above and shifting the Greenland polygon to Africa, the result looks like this:

It is now possible with the same method to shrink all countries worldwide to their "true-size". However, after using affine transformation, they will be mis-placed. So one has to shift them back.

This can be achieved by placing the centroid of the shrunk polygons back to the centroid of the original polygons (in Web mercator). For this, the x- and y-transformatation between the two centroids can be calculated: x-/y-coordinate value of the centroid in original Web Mercator layer minus x-/y-coordinate value of the centroid in the shrunk layer. This can be done with an expression like this one (adapting the layer and field-names) to calculate a field for x_transformation (and accordingly one for y_transformation as well):

x(
    centroid (
        geometry (
            get_feature (
                'world_mercator_3857', 
                'NAME', 
                "NAME"
            )
        )
    )
)
-
x(
    centroid (
        $geometry
    )
)

Then again use affine transformation, but this time only to shift the polygons, not scale them, using the x_transformation and y_transformation attributes calculated above. This is how the result looks like (compare with this: https://en.wikipedia.org/wiki/Mercator_projection#/media/File:Worlds_animate.gif):