Abstract Default reasoning and interpolation are two important forms of commonsense rule-based reasoning. The former allows us to draw conclusions from incompletely specified states, by making assumptions on normality, whereas the latter allows us to draw conclusions from states that are not explicitly covered by any of the available rules. Although both approaches have received considerable attention in the literature, it is at present not well understood how they can be combined to draw reasonable conclusions from incompletely specified states and incomplete rule bases. In this paper, we introduce an inference system for interpolating default rules, based on a geometric semantics in which normality is related to spatial density and interpolation is related to geometric betweenness. We view default rules and information on the betweenness of natural categories as particular types of constraints on qualitative representations of Ga?rdenfors conceptual spaces. We propose an axiomatization, extending the well-known System P, and show its soundness and completeness w.r.t. the proposed semantics. Subsequently, we explore how our extension of preferential reasoning can be further refined by adapting two classical approaches for handling the irrelevance problem in default reasoning: rational closure and conditional entailment.