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

22

21 sincos

)(sin)(cos 22

21

T

T

T

d1

d2

Principal directions: d and d . (1 unknown)1 2

Normal curvature in direction T :

Principal curvatures: and . (2 unknowns)1 2

)(sin)(cos 22

21

in direction T :

in direction T :

and can be controlled.

, , can be measured.

, , 1 2

tangent plane at p:

(closed forms)

?

d , d1 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:

TTN

( 1 – B N )

normal curvature in direction T

BT

normal to sampling plane

Track in two more planes through p.

Estimation over Synthetic Data

12

2

2

2

2

2

358 zyx

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

)(2

1 22

21 yxz

Local geometry approximated by an elliptic paraboloid:

p

too local!

Include higher order terms to represent a larger area:

ji

djiij yxayxz

3

22

21 )(

2

1

z

x

y

surface normal

principaldirections

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 4.

1. Track three curve segments intersecting at a reference point p.

Tangent interpolation to generate artificial surface points between the three curve segments.

ji

djiij yxayxz

3

22

21 )(

2

1

Preliminary Result: Top of a Sphere

Three views:

Patch on a Mouse

A different view: