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.
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
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
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.
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
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
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 Track three curve segments intersecting at a reference point p. Tangent interpolation to generate artificial surface points between the three curve segments.
Preliminary Result: Top of a Sphere Three views:
Patch on a Mouse A different view: