Intersections

A segment is uniquely defined by its two endpoints. By considering two line segments in plane geometry there are three possibilities:
the segments have zero points, one point, or infinitely many points in common.
Moreover when two segments have exactly one point in common it might be the case that that common point is an endpoint of either one of the segments or of both. If a common point of two segments is not an endpoint of either of the segments it is an interior point of both segments.
We will call a common point $T$ of two segments $L_1$ and $L_2$ a true intersection point of $L_1$ and $L_2$ if $T$ is the only common point of $L_1$ and $L_2$ and $T$ is an interior point of both segments.

Consider the three segments $L_1$, $L_2$, and $L_3$:

$L_1$: $(27, 44)$ to $(12, 32)$
$L_2$: $(46, 53)$ to $(17, 62)$
$L_3$: $(46, 70)$ to $(22, 40)$
It can be verified that line segments $L_2$ and $L_3$ have a true intersection point. We note that as the one of the end points of $L_3$: $(22,40)$ lies on $L_1$ this is not considered to be a true point of intersection. $L_1$ and $L_2$ have no common point. So among the three line segments, we find one true intersection point.
Now let us do the same for $5000$ line segments. To this end, we generate $20000$ numbers using the so-called “Blum Blum Shub” pseudo-random number generator.
\begin{align}
s_0 &= 290797\\
s_{n + 1} &= s_n \times s_n \pmod{50515093}\\
t_n &= s_n \pmod{500}
\end{align}
To create each line segment, we use four consecutive numbers $t_n$. That is, the first line segment is given by:
$(t_1, t_2)$ to $(t_3, t_4)$.
The first four numbers computed according to the above generator should be: $27$, $144$, $12$ and $232$. The first segment would thus be $(27,144)$ to $(12,232)$.
How many distinct true intersection points are found among the $5000$ line segments?

To solve this problem, we would need to iterate over every pair of line segments, calculate if they intersect, and if they do, check whether the point of intersection is a true intersection point. Steps to perform this operation would be:

1. Generate the 5000 line segments using the provided number generator.
2. For each pair of line segments, calculate their point of intersection if it exists. This is done by solving the system of equations that the two lines defined by the segments form.
3. Check if the point of intersection found in step 2, if it exists, is a true intersection point. This is done by checking if the point is not an endpoint for either of the segments and that it lies on both segments.

However, this problem is computationally demanding for manual calculations as it requires checking over 12 million pairs of segments. Hence, it is practically unsolvable by a human without assistance from a computer.

To automate this process, one could write a computer program in a language such as Python or JavaScript using available libraries for numerical computation and geometry. The essentials would be to:
– Generate the points for the line segments using the Blum Blum Shub generator.
– Create a function that will determine the intersection of two line segments
– Create a function that checks if the point is a “true intersection”
– Iterate over all combinations of the line segments and count the amount of “true intersections”.

Remember to use an appropriate data structure, like an ordered set or a list, to store the unique “true intersection” points such that there are no repetitions. Thus, providing a precise count of “distinct true intersection points.”

To create such a program, one must be proficient with programming and familiar with the associated mathematical concepts and their coding implementations, such as vectors for vector math, tuples for storing points, and arrays/lists for storing multiple items.

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