Three-dimensional (3D) medical imaging deals with the visualization, manipulation, and measuring of objects in 3D medical images. So far, research efforts have concentrated primarily on visualization, using well-developed methods from computer graphics. Very little has been achieved in developing techniques for manipulating medical objects, or for extracting quantitative measurements from them beyond volume calculation (by counting voxels), and computing distances and angles between manually located surface points. A major reason for the slow pace in the development of manipulation and quantification methods lies with the limitations of current algorithms for constructing surfaces from 3D solid objects. We show that current surface construction algorithms either (a) do not construct valid surface descriptions of solid objects or (b) produce surface representations that are not particularly suitable for anything other than visualization. We present ALLIGATOR, a new surface construction algorithm that produces valid, topologically connected surface representations of solid objects. We have developed a modeling system based on the surface representations created by ALLIGATOR that is suitable for developing algorithms to visualize, manipulate, and quantify 3D medical objects. Using this modeling system we have developed a method for efficiently computing principle curvatures and directions on surfaces. These measurements form the basis for a new metric system being developed for morphometrics. The modeling system is also being used in the development of systems for quantitative pre-surgical planning and surgical augmentation.