Axial representations of 3d shapes

Many shapes can be described very well in terms of an axis. Examples are the stick figure drawings of man and animals from ancient times. We would like to give a natural definition of axis for cylindrical shapes. For 2D shapes, the medial axis is a good candidate, though it is sensitive to perturbations. For 3D shapes, the medial scaffold is often complicated surfaces and does not match people's intuitions very well. In the first part of the thesis, we give a new definition of axis for generalized cylinders. This approach uses a regression point of view, defining the axis as a minimization point of a global energy function. We prove the existence of the minimization point and give the differential equations of it. We also apply the definition to some real and artificial examples. The second part of the thesis is about an application to potsherds, the symmetric axis and profile curve estimation of axially symmetric pots from small fragments. This problem is of great archaeological importance to the study of the hundreds and thousands of sherds found at excavation sites. The method is based on the special geometric properties of surfaces of revolution. We also use bootstrap method to find confidence bounds for the axis and the profile curve. These confidence bounds are essential if the estimations are used for the assembly of the full pot from multiple sherds.