The geometry of the Gauss product (Q1817430)

The product mentioned in the title is the composition of binary quadratic integral forms. In the case of definite forms, Gauss (apparently first in a review) gave a geometric representation in terms of lattices. \textit{F. Klein} [see Gesammelte Math. Abhandlungen. II (1973; Zbl 0269.01016), p. 299] gave a geometric interpretation of composition in terms of these lattices. The above were aware that one can compose forms whose discriminants differ by squares. In the present paper, the author arrives at a geometric description of such compositions, in terms of hypercycles (loci equidistant to a fixed geodesic) on the modular surface. The presentation is somewhat informal, with various intriguing parenthetical remarks attributed to the likes of Sullivan and Zagier. Although there is no clear application of the author's description, it seems reasonable to guess that Klein would have been pleased to see it.