We have devised and implemented in PrISM (protein informatics system for modeling) a new measure of protein structural relationships, the protein structural distance (PSD). The PSD is designed to describe relationships between protein structures in quantitative rather than descriptive terms and is applicable both when two structures are very similar, and when they are very different. It is calculated with a structural alignment procedure that uses double dynamic programming to align secondary structure elements and an iterative rigid body superposition that minimizes the root-mean-square deviation of C^α atoms. The alignment algorithm, as implemented on a modest workstation, is computationally efficient, allowing for large-scale structural comparisons. PSD scores for more than one and a half million pairs of proteins were calculated and compared to the discrete classification of proteins in the SCOP database. The PSD scores, which were obtained automatically, are in large part consistent with the manually derived classifications in SCOP. Discrepancies do arise, however, due, in part, to the fact that SCOP uses criteria other than structural similarity to derive classifications while the PrISM procedure is exclusively structure based. Analysis of PSD scores suggests that there is a continuous aspect of protein conformation space, even though various classification schemes are extremely useful. The use of a continuous measure for structural distance between all pairs of proteins allows us, as described in the two accompanying papers to derive sequence/structure relationships in a more quantitative way than has previously been possible. An important strength of the approach implemented in PrISM is its ability to address many different kinds of queries interactively, making its structural comparison procedure a convenient computational tool that complements structural classification databases such as SCOP and CATH.