The definition is : A conic is the locus of the intersections of corresponding lines in two projectively related pencils .
this is based on: for any four given points on a straight line appear under the same cross-ratio from any fifth point O - where in the case of a conic section the point is only free to move along the cone.
I have problems tying these two statements together because I can't grasp the concept of constructing it which is:
construct two points O & O' that have the property of biunique correspondence between the pencils of lines coming from them where: a line a coming from O meets a line a' coming from O' that lies on the cone.
ex. for projectively related pencils
The two pencils of lines in this state have the same cross ratio.
I'd love to be provided by a proof to that and\ or it's significance to the definition of the conic section along with definition \ explanation\ examples to biunique correspondence .
PS : My source is the book What is Mathematics from where the statements in bold are quoted.