How do straight lines look in this model of the projective plane? Suppose I take an Euclidean circle and identify antipodal points, then the inner points of the circle and its border are a model of the projective plane.
What would a straight line in this model mean?
How would a straight projective line in this model look like?
Or are there many alternatives?
 A: Think of the projective plane as half a sphere, border  included, with antipodal points in the border circle identified. A projective line is then half a great circle with the points in the border circle identified.
If you want to think of the "plane" model, think you are in R^3, project, say,  the unit sphere into the x,y-plane.Then you can think of the projective plane as being obtained, via orthogonal projection, from a unit disk with antipodal border points identified. Projective lines are then the projections of the half circles mentioned above with the border points identified.
Hope my english makes it clear.
A: "Straight" does not make sense since the construction you describe is a topological construction. If you draw a line on the disk, yes it will looks like a straight usual line of $\mathbb R^2$ intersected with $D^2$ (so the endpoint will be identified : topogically it will be a circle). 
But if you want to "imagine" what the lines on the projective space looks like, they are embedding on the projective space, and these embeddings will not be send lines of $\mathbb RP^2$ to straight lines.
