We report results from a systematic numerical investigation of the nonlinear patterns that emerge when a slender elastic rod is deployed onto a moving substrate; a system also known as the elastic sewing machine (ESM). The discrete elastic rods (DER) method is employed to quantitatively characterize the coiling patterns, and a comprehensive classification scheme is introduced based on their Fourier spectrum. Our analysis yields physical insight on both the length scales excited by the ESM, as well as the morphology of the patterns. The coiling process is then rationalized using a reduced geometric model (GM) for the evolution of the position and orientation of the contact point between the rod and the belt, as well as the curvature of the rod near contact. This geometric description reproduces almost all of the coiling patterns of the ESM and allows us to establish a unifying bridge between our elastic problem and the analogous patterns obtained when depositing a viscous thread onto a moving surface; a well-known system known as the fluid-mechanical sewing machine (FMSM).