First take a look at the blank figure in the lab book.
Then look at my new version below:
I've dashed the line that represents the celestial equator where it is behind the page. Then I've drawn in some geography. There's North America on the right with Alaska at the top an there you can see Baha and the Gulf of California. Madison is there on the edge just rotating out of view. In the mid Pacific is Hawaii and there's China and Japan on the left in the north. Australia and New Zeeland are at the lower left. Our Topical Island is there south of Baha, sort of where the Galapagos Islands are. Note where then the zeniths for the pole (it's at the pole), Madison and the Tropic Isl (on or at least near the equator) are. Next I've drawn in some meridian lines on the sky to highlight the idea that you are looking at the outside of a celestial sphere globe. Madison's meridian is the great circle that formes the outside of the globe. Finally I've sketched in a couple of familiar constellations which you might have noticed on the Voyager program, the Little Dipper and Orion. The motions of the stars you should have noticed are parallel to the celestial equator and go from right to left, i.e. westward. Thus Orion's belt would just skim along the equator while the bowl of the dipper would travel along "small" circles which would be parallel to the great circle of the equator.
Now remember that the HORIZON for any particular observer is a plane perpendicular to the line to the observers ZENITH. So now look below a the next diagram:
So first in BLUE are the motion arrows (we science geeks call them vectors). In red is the Horizon for Madison, the part that is behind the plane of the page is dashed. I've only shown the Tropical Isl. Horizon on the front side in green, but it is one of thoes meridians that goes from pole to pole (if the Island is right on the equator). And finally the celestial equator is the horizon at the pole. Now extend the diurnal star motion vectors to the various horizons They NEVER intersect the polar horizon! They hit the Madcity horizon at an angle of around 45 degrees and on our Tropic Isl. they hit the horizon perpendicular to it.
I sure hope this helps to visualize the situation we are trying to depict.