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A set $L$ of straight lines and a set $P$ of points in the Euclidean planedefine an arrangement $\mathcal{A}$ = ($L$, $P$) of construction lines andregistration marks, if and only if: (1) any point in $P$ is a point ofintersection of at least two lines in $L$, and (2) any two nonparallel lines in$L$ have a unique point of intersection in $P$. This paper discusses thefollowing open problems regarding such arrangements. Suppose $k \geq 0$ numberof points are given in the plane. How many construction lines $k$ points mustdetermine? How many distinct slopes, or directions, are defined by constructionlines that $k$ points determine? How many distinct sets of construction linespartition the plane, such that the lines meet at exactly $k$ points? Empiricalevidence is reported for small numbers of $k$, offering partial answers to thethree problems. A conjecture is also stated for the first problem, on thenumber of construction lines, after examining a related problem about finitelinear spaces from incidence geometry.

Some Open Problems Regarding the Number of Lines and Slopes in Arrangements that Determine Shapes