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?

• I don't understand how this gives you a model of the projective plane. If you have a line that does not pass through the center of the circle, then the two intersections with the border are not antipodal (and so represent two different points); then this line intersects the line at infinity (the border) in two points? – Morgan Rodgers Feb 1 '17 at 5:28
• No. The line you draw with two different points at infinity is only visually a line in te 2-D model om paper. It is not conceptually a line of the proj. plane. Therefore I asked: how does a straight line look like in the paper 2-D model. – Gerard Feb 2 '17 at 9:07

"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.
• This depends on the embedding of $\mathbb RP^2$ you will choose. For example, if you take a line in $\mathbb RP^2$, your line will be included in $\mathbb RP^2 \backslash D$ where $D$ is a disk, and since $\mathbb RP^2 \backslash D \cong M$ where $M$ is a Moebius band, this gives you straight line of $\mathbb RP^2$ as "soul" of a Moebius band, or as a boundary of a disk in $M$. Maybe you are interested in the curvature of the projective plane ? It is the same as the curvature of $S^2$. Geodesics of $\mathbb RP^2$ are projections of geodesics of $S^2$, so straight line in the disk model. – user171326 Jan 22 '17 at 11:19