Projective Geometry

What is the equation of this point?

• Find the line incident on these points in the projective plane.

• Find the point incident on both of these projective lines.

Find the formula for the line incident on two projective points. Find the formula for the point incident on two projective lines.

Prove that the definition of incidence is independent of the choice of the representatives of $p$ and $L$. That is, if $p_{1}$, $p_{2}$, $p_{3}$, and $q_{1}$, $q_{2}$, $q_{3}$ are two triples of homogeneous coordinates for $p$, and $L_{1}$, $L_{2}$, $L_{3}$, and $M_{1}$, $M_{2}$, $M_{3}$ are two triples of homogeneous coordinates for $L$, prove that $p_{1}L_{1}+p_{2}L_{2}+p_{3}L_{3}=0$ if and only if $q_{1}M_{1}+q_{2}M_{2}+q_{3}M_{3}=0$.

Give a drawing to show that central projection does not preserve circles, that a circle may project to an ellipse. Can a (non-circular) ellipse project to a circle?

Give the formula for the correspondence between the non-equatorial part of the antipodal modal of the projective plane, and the plane $z=1$.