We describe a novel continuous medial representation for object geometry and a deformable templates method for fitting the representation to images. Our representation simultaneously describes the boundary and medial loci of geometrical objects, always maintaining Blum's symmetric axis transform (SAT) relationship. Cubic b-splines define the continuous medial locus and the associated thickness field, which in turn generate the object boundary. We present geometrical properties of the representation and derive a set of constraints on the b-spline parameters. The 2D representation encompasses branching medial loci; the 3D version can model objects with a single medial surface, and the extension to branching medial surfaces is a subject of ongoing research. We present preliminary results of segmenting 2D and 3D medical images. The representation is ultimately intended for use in statistical shape analysis.