Conditions for shear and compaction localization are examined for stress states on an elliptic yield cap in the space of Mises equivalent shear stress $bar{tau}$ and mean normal compressive stress σ. Localization is predicted to occur when the slope of the hydrostatic stress versus inelastic volume strain curve falls to a critical value kcrit. When applied to the standard triaxial test, the analysis reveals a transition from compaction localization to shear localization as lateral confining stress σc decreases below a particular value. This value also corresponds to the largest value of kcrit, which is zero if normality is satisfied and slightly positive if not, for either shear or compaction bands. For σc larger than the transition value, compaction bands are the only mode of localized deformation predicted, and kcrit becomes increasingly negative with increasing σc. For σc smaller than the transition value, both shear bands and compaction bands are possible, but the value of kcrit is larger for shear bands. These results are consistent with experimental observations of compaction bands on relatively flat portions of the stress versus strain curve, corresponding to kcrit ≈ 0, and with their occurrence in a limited range of σc. As σc is decreased from the transition value, the predicted angle between the normal to the shear band and the maximum compression direction increases rapidly from 0¿ (for a compaction band) to 20¿--30¿. This rapid increase provides a possible explanation for the infrequent observation of very low angle shear bands.