Method
Sequential rotation correction across 3 axes: project onto a plane, find reference point pair A–B on the convex hull, then rotate to align. Repeat for each axis.
Reference point selection
Farthest hull point — partition the convex hull into 4 quadrants from the centroid, pick the farthest point in each quadrant, then select the pair with the greatest distance.
Restricted edge hull point — filter points exceeding 50% of both D_X and D_Y in each quadrant, pick the point closest to the axis in each quadrant, then select the pair with the greatest distance.
Minimum bounding rectangle — use rotating calipers to find the minimum bounding rectangle; the tooth orientation is determined by its long axis.
Correction steps
Z-axis twice (Steps 1-1 and 1-2 on the X-Y projection), then X-axis (Y-Z projection), then Y-axis (X-Z projection). The second Z rotation refines residual error after the first rotation changes the convex hull shape.
Tooth tilted in 3D space. No corrections applied yet.
Project onto the X-Y plane. Find reference points A and B on the convex hull. Rotate around the Z-axis to align A–B horizontally.
Re-project onto X-Y after the first Z rotation. The 3D rotation shifts points on the hull boundary, exposing a residual angular error. The second Z rotation corrects this remainder.
Project onto the Y-Z plane after both Z rotations. Find A and B, rotate around the X-axis to correct the lateral tilt.
Project onto the X-Z plane after the Z and X corrections. Find A and B, rotate around the Y-axis to correct the remaining anterior-posterior tilt.
Chen et al. project only the crown portion of the tooth. This illustration uses the full parametric tooth shape.