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We are studying difficult geometric problems in computer-aided mechanical design where visualization plays a key role. The research addresses the fundamental design task of contact analysis: deriving the part contacts and the ensuing motion constraints in a mechanical system. We have automated contact analysis of general planar systems via configuration space computation. Configuration space is a geometric representation of rigid-body interaction that encodes quantitative information, such as part motion paths, and qualitative information, such as system failure modes. The configuration space dimension equals the number of degrees of freedom in the system. Three-dimensional spaces are most important, but higher-dimensions are often useful. The qualitative aspects, which relate to the topology of the configuration space, are best understood by visualization. We explain what configuration space is, how it encodes contact information, and what research challenges it poses for visualization.