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This lecture covers the geometry of vault intersections, the supports resulting from these intersections, and various types of intersections, showcasing examples such as cloister vaults, crossed vaults, and ribbed vaults found in historical structures like the Diocletian Baths in Rome and Beauvais Cathedral. It also explores the generation of vaults, including crossed and fan vaults, with a focus on the architectural marvel of King's College Chapel in Cambridge. The discussion extends to the curvature of vaults and domes, distinguishing between single and double curvature, and the different types of supports required for these structures.