# A curve such that all lines on the plane intersect it : cont..

Further to this question (which appears more or less settled); "Is there a curve on plane such that any line on the plane meets it (a non zero ) finite times ?"

I ask now the upper bounds of the number of such intersections.....

Question Is there a curve on plane such that any line on the plane meets it atleast once but no more than k times ( with atleast one line intersecting k times) ?

The issue concerning the lowest achievable bounds for such intersections has also been discussed elsewhere

EDIT

• For the cases where upper bound is achieved I stipulate that such case be present at all points of the curve (except maybe for finitely many).

• We are also not forced to assume smoothness or even connectedness of the curve. (The answer as a set of points is here; through a non constructive proof)

• While "meeting" means intersection , Being tangent can be taken as a special case

• We also consider weather the constraints are satisfied as the curve moves towards $\infty$
• It's not clear to me what your edit means. Do you mean that all but finitely many lines intersect the curve at less than $k$ points? Commented Sep 22, 2013 at 5:09
• In every region of the curve there are lines which intersect it K times.
– ARi
Commented Sep 22, 2013 at 5:15
• The interesting modification is not to raise the maximum, but to raise the minimum from 1 to 2. That changes the problem greatly. Commented Sep 22, 2013 at 5:19
• The minimum is being discussed here
– ARi
Commented Sep 22, 2013 at 5:21
• @Ross That's not hard. Take rectangular parabolas $xy=1, xy=-1$ and add the 4 points $(-1,0), (1,0), (0,1), (0,-1)$. Unless you want connected? Commented Sep 22, 2013 at 5:21

If $k$ is odd and greater than 1, then the polynomial $\prod_{i=1}^k (x-i)$ will satisfy your conditions.

It is clear that any line will intersect this odd degree polynomial at least once. Any line will intersect this $k$ degree polynomial at most $k$ times. The line $y=0$ intersects this polynomial exactly $k$ times.

• Great. Just for analysis if we add the constraint where the cases for k times intersections are required to be infinitely many ? ( as against possibly one in your answer).
– ARi
Commented Sep 22, 2013 at 4:52
• @Ari There are infinitely many lines, of the form $y = \alpha$, where $\alpha$ is close to 0. Commented Sep 22, 2013 at 4:56
• Correct but majority of the curve is devoid of such cases, please see my Edit above
– ARi
Commented Sep 22, 2013 at 5:02
• "odd," $\:\mapsto\:$ "odd and greater than 1," $\;\;\;$
– user57159
Commented Sep 22, 2013 at 5:41
• As one sees there are intervals [a,b] where f(x) never gets intersected more than one times by a line.
– ARi
Commented Sep 23, 2013 at 16:34