- 03 Jun '08 15:30

My first instinct is to say no. I think I can prove that no continuous curve works, and clearly no bounded set works. But there could be some bizarre collection of discrete points that does the job - I can't see how to disprove that quite yet.*Originally posted by David113***I don't have a solution...**

Is there a set of points S in the plane, such that every straight line in the plane contains exactly 2 points of S? - 03 Jun '08 15:41

The closest thing that exists would probably be the Euclidean axes, which would work for all lines except y=0 and x=0*Originally posted by David113***I don't have a solution...**

Is there a set of points S in the plane, such that every straight line in the plane contains exactly 2 points of S? - 03 Jun '08 16:39
*What about the points that define a circle with a radius of infinity?*

That's the first thing that came to mind. It would have to be infinite, since if it were a finite number of points, one could easily draw a horizontal line far above the last point. I'm pretty sure that's the solution. - 03 Jun '08 16:52

A circumference is the immediate intuitive answer, but I fail to see what sense it makes to consider these intersections at infinity.*Originally posted by Jirakon**What about the points that define a circle with a radius of infinity?*

That's the first thing that came to mind. It would have to be infinite, since if it were a finite number of points, one could easily draw a horizontal line far above the last point. I'm pretty sure that's the solution.

Give me a functional form taking the limit of r to infinity and I'll give you an epsilon that proves that there is at least a line that doesn't intersect on two points. - 03 Jun '08 17:19

Care to mention one of the points in that set then?*Originally posted by Jirakon**What about the points that define a circle with a radius of infinity?*

That's the first thing that came to mind. It would have to be infinite, since if it were a finite number of points, one could easily draw a horizontal line far above the last point. I'm pretty sure that's the solution. - 03 Jun '08 17:54

A point is an ordered pair of two REAL numbers. No infinity.*Originally posted by FabianFnas***What about the points that define a circle with a radius of infinity?**

Then every line should intersect this circle in exactly two points, at the polar coordinates r1=r2=inf, and phi1=f and phi2=f+pi.

Is this the answer? - 03 Jun '08 18:01 / 3 edits

I don't think it can be discrete points. Imagine if you have a set of discrete points and pick a line that intersects only two points. If you pick one of those points as center and keep rotating that line infinitesimally, then I don't see how it's possible to cover all possibilities with any mesh of discrete points.*Originally posted by mtthw***My first instinct is to say no. I think I can prove that no continuous curve works, and clearly no bounded set works. But there could be some bizarre collection of discrete points that does the job - I can't see how to disprove that quite yet.**

Edit - So I'd say no, there is no such set S.

Edit 2 - Which probably means I'm wrong and missing an ingenious answer. - 03 Jun '08 21:26
*Originally posted by Palynka***A circumference is the immediate intuitive answer, but I fail to see what sense it makes to consider these intersections at infinity.**

.

if you do include any trans finite numbers in the plane; then ithink that you need to include all trans finite numvbers.

another quwstion is: can you define a plane, and lines on that plane, so that set s exists. - 03 Jun '08 21:33
*A point is an ordered pair of two REAL numbers. No infinity.*

If that's the case, then there is no solution:

Suppose such a set exists in which all points have finite coordinates. Take the point(s) with the highest y-value (y_max). Now imagine the line y = y_max + 1. This line does not intersect any of the points. Therefore no such set exists. - 03 Jun '08 21:47

Well, it was a wild guess, the closest I can get to an answer.*Originally posted by Jirakon**What about the points that define a circle with a radius of infinity?*

That's the first thing that came to mind. It would have to be infinite, since if it were a finite number of points, one could easily draw a horizontal line far above the last point. I'm pretty sure that's the solution.

Is it possible to construct a plane where inf really is included? As you can with R, i.e. R* ?