# Understanding the phrase “Nondegenerate conics in general position” in the context of the real projective plane

I am reading an article about conics and one of the theorems states: "Let $$K$$ and $$C$$ be nondegenerate conics in general position..."

I am trying to apply this to real projective plane and I have trouble understanding this. General position means no three points are collinear, hence when talking about conic, it does not include a line. If a conic includes line, it is degenerate. So do we have to assume both things or is nondegenerate conics enough? Or do we have to say that $$K$$ and $$C$$ are nondegenerate and not empty (and we can forget about general position)? I assume in other projective planes things are more complicated.

• I think that general position signifies that the center of the conic (if it exists) is arbitrary, not necessarily $(0,0)$ ... and axis of symmetry is not necessarily parallel to a coordinates axis. Nondegenerate conic is a circle, ellipse, hyperbola or parabola (it is not empty, is not a single point, a line, a pair of lines). – user376343 Aug 22 at 21:30
• The phrase "general position" means a lot of different things in different contexts. Perhaps OP could provide more context. – Nate Aug 22 at 21:32
• To say that the two conics are in general position should signify that they do not intersect tangentially. Since they're nondegenerate, they cannot have a component in common, so that case is already ruled out. – Ted Shifrin Aug 22 at 22:05
• @tedshirfin to not intersect tangentially, does that equals they don't have a common tangent? – MocS Aug 22 at 22:07

For example, if you're examining the incidence structure of affine subspaces of $$\mathbb R^n$$, then a set of points could be called in general position iff for all $$k$$, no $$k+2$$ points lie in a $$k$$-dimensional subspace. If you're examining rational polynomials, a set of real or complex numbers could be called in general position if they are algebraically independent, i.e. there is no nontrivial polynomial equation involving these numbers and rational coefficients.