We describe a system that builds quantitative structural descriptions of spiral galaxies. This enables translation of sky survey images into data needed to help address fundamental astrophysical questions such as the origin of spiral structure - a phenomenon that has eluded full theoretical description despite 150 years of study. The difficulty of automated measurement is underscored by the fact that, to date, only manually-guided efforts (such as the citizen science project Galaxy Zoo) have been able to extract structural information about spiral galaxies. An automated approach is needed to eliminate measurement subjectivity and handle the otherwise-overwhelming image quantities (up to billions of images) from near-future surveys. Our approach automatically describes spiral galaxy structure as a set of arcs fit to pixel clusters, precisely characterizing spiral arm segment arrangement while retaining the flexibility needed to accommodate the observed wide variety of spiral galaxy structure. The largest existing quantitative measurements were manually-guided and encompassed fewer than 100 galaxies, while we have already applied our method to nearly 30,000 galaxies. Our output is consistent with previous information, both quantitatively over small existing samples, and qualitatively with human classifications.