I started learning projective geometry recently and i know just the basics how cross ratios are defined and some applications as given in Alexander Remorov's projective geometry note. But recently i came across a note which is titled "The method of moving points" and i see this method frequently used in solving geometry problems at Art of Problem Solving forum. But i am facing difficulties in understanding this method mainly because of the Notations. I have always been a bit uncomfortable with function symbols and notations in geometry branch of mathematics. But i know how to do everything in euclidean geometry way and i believe i am an amateur in euclidean geometry problem solving.
Here in the above picture in the first point, they said that a map from $l$ to $C_p$ is given by $X \rightarrow PX$ is projective where $l$ is a given line and $P$ is a given point and $C_p$ means pencil of lines through P. I don't understand what they are trying to mean. I know that pencil through $P$ means some lines are passing through $P$ but don't understand what map from $l$ to $C_p$ means. The projective map definition they gave was that from a conic $C_1$ to $C_2$ if $A,B,C,D$ are four points on $C_1$ and $f(A),f(B),f(C),f(D)$ are their images then $(A,B;C,D)=(f(A),f(B);f(C),f(D)$. But i don't understand what this map means since i said i am not very comfortable with function terminologies in geometry although o learnt those in Algebra.
Also they said inversion of a line is a projective map.
Could you please translate them and prove them in simple terms and euclidean way so that i can understand what they are trying to mean?That will be a very big help.