We present a geometric approach for constructing shape atlases of sulcal curves on the human cortex. Sulci and gyri are represented as continuous open curves in R3, and their shapes are studied as elements of an infinite-dimensional sphere. This shape manifold has some nice properties--it is equipped with a Riemannian L2 metric on the tangent space and facilitates computational analyses and correspondences between sulcal shapes. Sulcal mapping is achieved by computing geodesics in the quotient space of shapes modulo rigid rotations and reparameterizations. The resulting sulcal shape atlas is shown to preserve important local geometry inherently present in the sample population. This is demonstrated in our experimental results for deep brain sulci, where we integrate the elastic shape model into surface registration framework for a population of 69 healthy young adult subjects.