We present a method that maps a complex surface geometry to an equally complicated, similar surface. One main objective of our effort is to develop technology for automatically transferring surface annotations from an atlas brain to a subject brain. While macroscopic regions of brain surfaces often correspond, the detailed surface geometry of corresponding areas can vary greatly. We have developed a method that simplifies a subject brain's surface forming an abstract yet spatially descriptive point cloud representation, which we can match to the abstract point cloud representation of the atlas brain using an approach that iteratively improves the correspondence of points. The generation of the point cloud from the original surface is based on surface smoothing, surface simplification, surface classification with respect to curvature estimates, and clustering of uniformly classified regions. Segment mapping is based on spatial partitioning, principal component analysis, rigid affine transformation, and warping based on the thin-plate spline (TPS) method. The result is a mapping between topological components of the input surfaces allowing for transfer of annotations.