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.

  • $\begingroup$ 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). $\endgroup$ – user376343 Aug 22 at 21:30
  • $\begingroup$ The phrase "general position" means a lot of different things in different contexts. Perhaps OP could provide more context. $\endgroup$ – Nate Aug 22 at 21:32
  • $\begingroup$ 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. $\endgroup$ – Ted Shifrin Aug 22 at 22:05
  • $\begingroup$ @tedshirfin to not intersect tangentially, does that equals they don't have a common tangent? $\endgroup$ – MocS Aug 22 at 22:07

"General position" is a term that means much more than that three points are not collinear. Something is said to be in general position if it is not in any special position, i.e. anything you do with it will not result in any special case for the purposes of whatever you're doing.

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.

You see I'm using the word "could", because there's not really a formal definition of "general position", and how many special cases you want to exclude always depends on what you're doing.

In your case of two conics, a simple special case that would usually be excluded by "general position" would be the case where the two conics touch.

TL;DR: Just think of "X is in general position" as "X does not cause any special cases that could interfere with the following reasoning".


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