Differential Geometry
Some figures from Cipolla & Giblin and DeCarlo et al.
Goal
• Analyze qualitative shape of surface
(e.g., convex vs. concave vs. saddle)
• Understand surface curvatures
• Approach: parameterize surface, look at
derivatives in different directions
Surface Parameterization
• Locally smooth, continuous parameterization
(u,v)  s(u,v) = [X(u,v), Y(u,v), Z(u,v)]T
• Consider only on local patches of the surface
Surface Parameterization
• Locally smooth, continuous parameterization
(u,v)  s(u,v) = [X(u,v), Y(u,v), Z(u,v)]T
• Consider only on local patches of the surface
• r defines an “immersed surface” iff Jacobian has
rank 2 (otherwise locally looks like a line)
J  su
 Xu

sv    Yu
Z
 u
Xv 

Yv 
Z v 
Tangent Planes and Normals
• The vectors su and sv define a tangent plane
• Defined iff su and sv are not parallel
– Equivalent to saying s is an immersed surface
• Unit surface normal =
normal to tangent plane
su  sv
n
su  sv
n
sv
su
Vectors on Surfaces
• Consider two vectors w1 and w2 defined at a
point p on the surface
• Basic operation: how to compute dot product?
• Trivial if vectors defined in terms of (x,y,z)
• What if defined in terms of (u,v)? That is,
 w1,u 
 w 2,u 
, w 2  

w1  
 w1,v 
 w 2 ,v 
• In general, w1  w 2  w1,u w 2,u  w1,v w 2,v
First Fundamental Form
w1  w 2  w1,u su  w1,v sv   w 2,u su  w 2,v sv 
 su  su
 w1,u w1,v 
 su  sv
 I (w1 , w1 )
su  sv  w 2,u 


sv  sv  w 2,v 
• I is called the first fundamental form
– Symmetric bilinear form on the surface
– Specific to a particular parameterization
– Matrix is identity iff su and sv are orthonormal
Curves on Surfaces
• Consider a curve c(t) = (U(t), V(t))
• Velocity along the curve
c(t )  suU (t )  svV (t )
2
 I (U ,V )
• So, I describes how much surface is “stretched”
• Does not change if surface is “bent”
without stretching
Areas on Surfaces
• For a region on a surface, area is given by
 s  s dudv
   s s sin  dudv
A
u
u
v
v



( su  sv ) 2 
 dudv
( su  su )( sv  sv )1 
 ( su  su )( sv  sv ) 


det( I ) dudv
• So, sqrt of det of matrix I gives stretch of area
Normal Curves
• A normal curve is defined by the intersection of
a surface and a plane containing the normal
Normal Curvatures
• Curvature  of a curve is reciprocal of radius of
circle that best approximates it
• Curvature of a normal curve is called a
normal or sectional curvature
• Defined at a point p in a particular direction w
Finding Normal Curvatures
• Consider a normal curve c(t) = (U(t), V(t))
n
n+dn
c(t)
r
r
c
n 
c
c
r
n  c
 
I(c, c)
Second Fundamental Form
• So, need to find component of change in
normal in the c’ direction as we move along
curve (i.e., in the c’ direction)
• More generally, can ask about change in normal
in direction w1 as we move in direction w2
• This is called the second fundamental form
II(w1, w2)
– Also a symmetric bilinear form on the surface
Second Fundamental Form
II (w1 , w 2 )  w1  Dw 2 (n)
 w1  nu (w 2  su )  n v (w 2  sv ) 
 w1,u
 su  n u
w1,v 
 su  n v
sv  nu  w 2,u 


sv  nv  w 2,v 
• Note sign convention – this produces positive
curvature for convex surfaces with outwardpointing normals
II and Second Derivatives
su  n  0
suu  n  su  nu  0
su  nu  suu  n
 su  n u

 su  n v
sv  nu 
 suu  n suv  n 
  

sv  nv 
 suv  n svv  n 
• So II is related to the second derivatives
of the surface
Normal Curvatures
• The normal curvature in the direction w can be
written as
II (w, w )
w 
I (w, w )
Principal Curvatures
• The normal curvature varies between some
minimum and maximum – these are the
principal curvatures 1 and 2
• These occur in the principal directions d1 and d2
• Eigenvalues and eigenvectors of the matrix I-1II
– 1 and 2 must exist and be real
(symmetric matrix)
– d1 and d2 must be
perpendicular to each other
Frénet Frames
• d1, d2, and n are mutually perpendicular
• Natural to define a coordinate system according
to these directions – the Frénet frame
• In this coordinate system, the matrix of II
becomes diagonal
Euler’s Formula
• Working in the Frénet frame,
w
w 
d1
  1 0  w d1 


w d 2 

w
0

d2 
2 

2
w

  1 cos2    2 sin 2 
• Euler’s formula for normal curvature
•  is the angle between w and d1
Computing Curvatures on Meshes
• For range images: use (u,v) parameterization of
range image, finite differences for derivatives
• Options for general meshes:
– Fit a smooth surface, compute analytically
– Compute normal curvature to neighboring vertices,
least squares fit to a quadratic in 
– Compute normal curvature to neighboring vertices,
do SVD (curvatures are some function of singular
values)
Gaussian and Mean Curvature
• The Gaussian curvature K = 1 2
• The mean curvature H = ½ (1 + 2)
• Equal to the determinant and half the trace,
respectively, of the curvature matrix
• Enable qualitative classification of surfaces
Positive Gaussian Curvature:
Elliptic Points
• Convex/concave depending on sign of H
• Tangent plane intersects surface at 1 point
Negative Gaussian Curvature:
Hyperbolic Points
• Tangent plane intersects surface along 2 curves
Zero Gaussian Curvature:
Parabolic Points
• Tangent plane intersects surface along 1 curve
Properties of Curvatures
• Attached shadows always created along
parabolic lines
• Sign of curvature of image contours (silhouettes)
equals sign of Gaussian curvature of surface
– Inflections of the contour occur at parabolic points
on the surface
• Apparent motion of specular highlights ~ 1/
Using Curvatures for Recognition/Matching
• Curvature histograms: compute 1 and 2
throughout surface, create 2D histograms
• Invariant to translation, rotation
• Alternative: use 2 / 1 – also invariant to scale
– Shape index: S  12  1 tan 1    [0..1]
1
2
1
2
• Curvatures sensitive to noise (2nd derivative…),
so sometimes just use sign of curvatures
Using Curvatures for Segmentation
• Sharp creases in surface (i.e., where |1|is large)
tend to be good places to segment
• Option #1: look for maxima of curvature in the
first principal direction
– Much like Canny edge detection
– Nonmaximum suppression, hysteresis thresholding
• Option #2: optimize for both high curvature
and smoothness using graph cuts, snakes, etc.
Using Curvatures for Visualization
• Ridge and valley lines often drawn
– Local maxima and minima of larger (in magnitude)
curvature in principal direction
– Makes interpretation of surfaces easier in some cases
• Suggestive contours
– Zeros of normal curvature in projection of view
direction into tangent plane
– Positive derivative of curvature in projected view dir