consider the set of lines defined by all pairs of points $\lbrace[(u,v),(u-v,v+u)],\ u,v\in\mathbb{N}\rbrace$ in the euclidean plane
Question:
what is kown about the density of the set of intersection-points of pairs of non-identical lines from the set of lines as defined above
- is this set of intersection points dense in the first quadrant
- what can be said about their density/distribution
is this set of intersection points dense in the first quadrant
No. Indeed, let $\ell_{u,v}$ be the line through the points $(u,v),(u-v,v+u)$. The point of intersection of distinct nonparallel lines $\ell_{u,v}$ and $\ell_{x,y}$ is $$p_{u,v,x,y}:= \frac1{uy-vx}\,\big((u^2+v^2)y-(x^2+y^2)v,(x^2+y^2)u-(u^2+v^2)x\big),$$ with $uy-vx\ne0$.
Proposition 1: For any real $a>0$, $b>0$, $u\ge0$, $v\ge0$, $x\ge0$, and $y\ge0$ such that $p_{u,v,x,y}=(a,b)$, one must have $u\le a+b$ and $v\le a+b$ and hence, by symmetry, $x\le a+b$ and $y\le a+b$.
See the proof of Proposition 1 below.
By Proposition 1, for any real $R>0$, the set $S_R:=\{p_{u,v,x,y}\in(0,R]^2\colon u,v,x,y\text{ in }\Bbb N\}$ is of cardinality $\le(R+R+1)^4<\infty$, so that the set $S:=\{p_{u,v,x,y}\colon u,v,x,y\text{ in }\Bbb N\}$ is not dense in $(0,R]^2$. In fact, being locally finite, the set $S$ is nowhere dense in the first quadrant. $\quad\Box$
Proof of Proposition 1: This is a problem of real algebraic geometry, which therefore admits a purely algorithmic solution. Such algorithms are realized in Mathematica by commands such as Reduce. Using this command, in about 0.12 sec we get
$\Box$
Proposition 1 can also be proved by hand, by considering four rather simple cases to prove that $u\le a+b$, depending on the signs of $uy-vx$ and $u-v$, and similar four cases to prove that $v\le a+b$, and using certain convexity and monotonicity patterns. That proof is quite elementary but long and tedious.
