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I am posting here because I need help finding (or making) a visual aid for a presentation.

I am giving a short presentation about Projective Geometry next week, and I am building a beamer for it. One such slide talks about the Standard Lift of the Projective Plane, and how it serves as a way to coordinatize the Plane (albeit with infinitely many different coordinate triples for any point in the Projective Plane).

I am trying to find a diagram of the Standard Lift of the plane. I.e. define $\mathbb{RP} ^2$ as the set of lines through the origin in $\mathbb{R}^3$, and then consider the plane $z=1$. Each line (except those sitting in the $xy$-plane) intersects the plane $z=1$ at exactly one point, $(x,y,1)$ and we can use that as a representative vector of the point $[x,y,1] \in \mathbb{RP} ^2$. Furthermore, the lines sitting in the $xy$-plane form the ideal line of $\mathbb{RP} ^2$.

Does anyone know where I could find a diagram of what I just described? Maybe one of Euclidean 3-space with axes labeled, the plane $z=1$, and a few examples? Or have suggestions of software I can use to easily draw this? Preferably I could just use an existing diagram but I know that might be wishful thinking.

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If you google images of projective plane, you might find what you want. For example: here. Latex and Matlab have some tools for drawing.

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