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The problem of geometric (model and system) interoperability is a non-trivial generalization of the classical problem of part interchangeability in mechanical assemblies. Interoperability subsumes the challenging problems of geometric model quality, exchange, and interchangeability, as well as system integration. Until now, most of the interoperability solutions adopted a data-centric approach that requires exchange and translation of standardized file formats and/or computer representations. The difficulties with a data-centric approach include lack of consistent supporting mathematical models, high overhead in supporting new representations, and inability to provide local, adaptable, and incremental interoperability solutions. We adopt an alternative approach to interoperability using queries: formally specified computable functions that are widely supported by different systems. We propose a hierarchy of such queries and demonstrate that they are adequate to provide interoperable solutions to many common geometric tasks in computer aided design and manufacturing, including model acquisition and exchange, metrology, and also computer aided design/analysis integration.