You can tell where on the circle to put the points because the latitude of the location tells you how far up or down you need to put the point.

How?

Well you agree that latitudes and longitudes are correct for both models?

If it is sunrise at a certain point and sunset at another, you can (using the latitudes) figure out where on the edge of the circle they are because the point will intersect the Spotlight area.

You would also have to take into account the time of year (How far North/South the Sun is compared to Equator) to get a better approximation.

In keeping the Spotlight the same size, it would eventually scale itself out.