We define a shape as a union of finitely many line segments. Given an arrangement of lines on a plane, we count the number of shapes in the arrangement by examining the symmetries of the arrangement and applying Burnside's lemma. We further establish a generating function for the number of distinct line segments on a line with k distinguished points. We list all affine line arrangements of four and five line segments, together with the corresponding number of shapes on them.
Dr. Josephine Yu
Cai, May and Liao, Nicholas
"On the Enumeration of Shapes,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 21
, Article 3.
Available at: https://scholar.rose-hulman.edu/rhumj/vol21/iss1/3