We describe the HIPAIR configuration space computation program for higher pairs and show how it automates reasoning about shape and motion for mechanism design. We describe an interactive parametric design module that combines configuration space computation with differential constraint satisfaction. HIPAIR handles pairs of 2.5D parts with two degrees of freedom, including pairs with intermittent, simultaneous, and degenerate contacts. This class contains 90% of 2.5D pairs and 80% of all higher pairs according to our survey of 2500 mechanisms. We have tested HIPAIR on over 100 pairs, including gears, cams, ratchets, and escapements. It analyzes pairs with thousands of contacts in under ten seconds. The configuration spaces encode the relations among part shapes, part motions, and overall behavior in a concise, complete, and explicit format. They help designers analyze part interactions, implement functions, identify failure modes, and modify designs.