How can I show the relative size of two GeoGraphics objects?

Question

I've created a GeoGraphics object from data I collected on a recent trip across part of Michigan, like so:

I'd really like to overlay another country to scale in the center of this image to give a more tangible measure of the length of the path, but I'm running into trouble keeping everything lined up correctly. If I do something like

I just get the portion of the globe that encompasses both my path and the UK:

How can I "move" the to-scale outline of the UK over the outline of my path?

Answer 1

Due to the non-sphericity of the Earth, there is no exact way of rotating a polygon, but the following is a good approximation.

Suppose we want to rotate the UK polygon so that London is moved to Lansing:

In[]:= london = GeoPosition[Entity["City", {"London", "GreaterLondon", "UnitedKingdom"}]];

In[]:= lansing = GeoPosition[Entity["City", {"Lansing", "Michigan", "UnitedStates"}]];

That means we need to implement these changes in latitude and longitude:

In[]:= dlat = Latitude[lansing] - Latitude[london]
Out[]= Quantity[-8.79021, "AngularDegrees"]

In[]:= dlon = Longitude[lansing] - Longitude[london]
Out[]= Quantity[-84.4396, "AngularDegrees"]

We will do it in two steps, first changing latitudes along the London meridian and then changing longitudes along the Lansing parallel. This will preserve the North-orientation of the UK.

In[]:= axis = FromSphericalCoordinates[{1, Pi/2, -Pi/2 + Normal@Longitude[london]}];
Out[]= {-0.00203622, -0.999998, 0};

In[]:= rot = Transpose[RotationMatrix[dlon, {0, 0, 1}].RotationMatrix[dlat, axis]];

Get the points of our polygon of the UK:

In[]:= UK = First@EntityValue[Entity["Country", "UnitedKingdom"], "Polygon"];

Finally perform the rotation (note that we act with the rot matrix on the right, which explains the use of Transpose before):

In[]:= newUK = GeoPosition@ GeoPositionXYZ[Dot[#, rot] & /@ First@GeoPositionXYZ[UK]];

Now we can plot:

In[]:= GeoGraphics[{GeoMarker[lansing], Polygon[newUK]}]