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This paper deals with the problem of precise automatic estimation of the surface geometry of pot sherds uncovered at archaeological excavation sites using dense 3D laser-scan data. Critical to ceramic fragment analysis is the ability to geometrically classify excavated sherds, and, if possible, reconstruct the original pots using the sherd fragments. To do this, archaelogists must estimate the pot geometry in terms of an axis and associated profile curve from the discovered fragments. In this paper, we discuss an automatic method for accurately estimating an axis/profile curve pair for each archeological sherd (even when they are small) based on axially symmetric implicit polynomial surface models. Our method estimates the axis/profile curve for a sherd by finding the axially symmetric algebraic surface which best fits the measured set of dense 3D points and associated normals. We note that this method will work on 3D point data alone and does not require any local surface computations such as differentiation. Axis/profile curve estimates are accompanied by a detailed statistical error analysis. Estimation and error analysis are illustrated with application to a number of sherds. These fragments, excavated from Petra, Jordan, are chosen as exemplars of the families of geometrically diverse sherds commonly found on an archeological excavation site. We then briefly discuss how the estimation results may be integrated into a larger pot reconstruction program.