What is curvature

1
What is curvature
Small movements along the surface ? change in
surface normal
n
2
Vector calculus

• Taylor expansion of scalar field f

isosurface f(x) C
x0
? Hessian is basis of curvature
3
Projections, Hessian, Invariants

• Consider

n
4
Projections, Hessian, Invariants

• Consider

• nnT

nnT
5
Projections, Hessian, Invariants

• Consider

P I - nnT

• nnT
• P I - nnT

6
Projections, Hessian, Invariants

• Consider

• nnT
• P I - nnT
• H

H
7
Projections, Hessian, Invariants

• Consider

• nnT
• P I - nnT
• H
• HP

HP
8
Projections, Hessian, Invariants

• Consider

• nnT
• P I - nnT
• H
• HP
• PHP

PHP
9
Projections, Hessian, Invariants

• Consider

• nnT
• P I - nnT
• H
• HP
• PHP

PHP
10
Projections, Hessian, Invariants

• Consider

G -PHP/g

• nnT
• P I - nnT
• H
• HP
• PHP
• G -PHP/g

11
Projections, Hessian, Invariants

• Consider

G -PHP/g

• nnT
• P I - nnT
• H
• HP
• PHP
• G -PHP/g