Curvature
constraints from Gauss Bonnet theorem
gaussian curvature cg ¼ 1/(R1 R2)
R2
R1
Gauss Bonnet theorem:
˜
cg ds + ˜ cp dl
surface
boundary
does not change when
surface is deformed
cp is Ògeodesic curvatureÓ––the
curvature as drawn on the surface.
does not change when
flat sheet is deformed into a d-cone*
* unless the boundary stretches
Thus ˜ cg ds ¨ 0
h¨0
So, rim gaussian
curvature not clearly related to core gaussian curvature
large positive and
negative regions ~ cancel