The use of conics in projective geometry

Question

What is the use of conics in projective geometry. We dealt with it in one of my lectures and defined conics and my lecturer said it is an imprtant kind of geometric object in projective geometry. But i don't know the use of conics in the projective geometry. Why is it important?

Answer 1

Colloquially speaking, conics are to projective geometry what circles are to Euclidean geometry.

A circle is not a projective concept: applying a projective transformation to a circle will in general transform it into a non-circular conic. So the class of all conics, which is closed under projective transformations (i.e. applying a projective transformation to a conic yields another conic) is the most reasonable generalization of a circle in this context. (In this light, a circle is a special conic which passes through the two ideal complex-valued circle points $(1,\pm i,0)$.)

There are many situations where conics arise naturally in projective geometry. For example, the set of all points $X$ for which the cross ratio $(A,B;C,D)_X$ (i.e. the cross ratio of the four lines connecting $X$ to $A,B,C,D$ in turn) is some arbitrary constant will be a conic section. Some properties related to conics, like e.g. the pole-polar relationship, are also very natural concepts in projective geometry since they can be characterized by incidence relations without any need for measurements.