A novel and efficient invertible transform for shape segmentation is defined that serves to localize and extract shape characteristics. This transform -- the chordal axis transform (CAT) -- remedies the deficiencies of the well-known medial axis transform (MAT). The CAT is applicable to shapes with discretized boundaries without restriction on the sparsity or regularity of the discretization. Using Delaunay triangulations of shape interiors, the CAT induces structural segmentation of shapes into limb and torso chain complexes of triangles. This enables the localization, extraction, and characterization of the morphological features of shapes. It also yields a pruning scheme for excising morphologically insignificant features and simplifying shape boundaries and descriptions. Furthermore, it enables the explicit characterization and exhaustive enumeration of primary, semantically salient, shape features. Finally, a process to characterize and represent a shape in terms of its morphological features is presented. This results in the migration of a shape from its affine description to an invariant, and semantically salient feature-based representation in the form of attributed planar graphs. The research described here is part of a larger effort aimed at automating image understanding and computer vision tasks.