<<Axis/Profile-Curve Figures>> click image to see large

Each of rows 1 through 3 illustrates a different sherd, the estimated axis, and the estimated profile curve.  For these examples, the axis/profile-curve pair is estimated by fitting a 3D algebraic surface, restricted to be rotationally symmetric around an axis, to the 3D data.  The degree of the algebraic surface used for a particular sherd is the degree that is appropriate to the complexity of the shape.  The degree is chosen automatically, and I believe it ranges from 2 to 6? for the three examples shown.  The first-two columns for a row illustrate views of the sherd outer-surface measurement data and the associated estimated axis.  The estimated axis is in yellow.  The third column is a view of the 3D measurement data and the 3D points on the estimated sherd surface rotated about the estimated axis into a plane.  The rotated points on the estimated 3D surface constitutes the estimated profile curve.  The profile curve is in blue, the data is in yellow. We an estimated axis and an estimated profile curve for the entire pot, but only Andrew knows where that is, so we will not know until the end of June.  Also, the pot is not perfectly symmetric about its axis.  One can observe local variations.

Row 1: a fragment of the side and the lip.  Accurate estimates of the axis and the profile curve because in the horizontal plane of the pot, i.e., the plane perpendicular to the axis, the extent of the sherd is large enough to measure the curvature, equivalently, the pot radius, and the profile curve is distinctive.

Row 2: a fragment of the side -- long in the vertical direction, short in the horrizontal direction.  Fairly accurate axis estimation, good profile-curve estimation.  In the second column, one can see that in the bottom right-hand corner, the side curves outward.  This accounts for the cluster of points at the bottom of the profile curve in the third column.

Row 3: a very small fragment of the side.  The fragment looks like a patch of a spherical surface, and the estimated axis is perpendicular to the sherd which is incorrect.  The profile curve fits the data well.  In practice, our algorithm gets a fast initial linear least-squares estimate of the axis through use of Plucker coodinates.  The estimator returns three estimates.  The minimum-cost estimate is not always the correct one, so the algorithm then uses each of the three as a starting point for the more accurate nonlinear estimator.  One of those final three estimates is usually the correct one, and can be determined automatically. In this case, Andrew just left us with the least-cost Plucker solution, so I do not know whther one of the others would give us a good estimate -- it may well not have. If a sherd is small, it is often the case that it looks like a patch of a sphere, and it may then be possible to estimate principal curvatures but not the axis for the sherd.

   sherd # 1

   sherd # 9

    sherd # 12

  click image to see large

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