1. Normal Curvature of Surface p  N T Local geometry at a surface point p:  surface normal N. The plane containing N and T cuts out a curve  on the surface.  tangent direction T. Normal curvature in T  curvature of  surface bending along T normal plane Every tangent direction at p defines a normal curvature.

2. Principal Curvatures tangent plane p N  Normal curvature function generally has unique minimum and maximum.  These extrema are achieved at two orthogonal tangent directions at p. principal directions principal directions principal curvatures  and  1 2

3. Solution of Principal Curvatures T  T  T  d 1 d 2       Principal directions: d and d. (1 unknown) 12 Normal curvature in direction T :   Principal curvatures:  and . (2 unknowns) 12 in direction T :     and  can be controlled.   , ,  can be measured.    , ,  1 2  tangent plane at p: (closed forms) ? d, d 1 2

4. Measuring Normal Curvature  normal plane N sampling plane Track the shape with a touch sensor constrained in a sampling plane through p. (planar contour tracking) Curve segments , , . p Tangents at p: T, T, T    Curvatures: , ,     Surface normal:   ( 1 – B  N ) normal curvature in direction T  B  T  normal to sampling plane Track in two more planes through p.

5. Estimation over Synthetic Data y = 1  2  1 z x y (0.047,0.125) (0.0284, 0.798) exact principal curvatures estimates (0,1,0)  2  1

6. Patch Reconstruction Local geometry approximated by an elliptic paraboloid: p  too local! Include higher order terms to represent a larger area: z x y surface normal principal directions Darboux frame determined by fitting

7. Reconstruction Algorithm 2. Fit over the curve segments in local coordinates.  Planar contour tracking (Part III)  Estimate principal curvatures and axes (Part I) Ongoing work …  Order of polynomials set to be 4. 1. Track three curve segments intersecting at a reference point p.  Tangent interpolation to generate artificial surface points between the three curve segments.

8. Preliminary Result: Top of a Sphere Three views:

9. Patch on a Mouse A different view: