1. Differential Geometry
Some figures from Cipolla & Giblin and DeCarlo et al.

2. 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

3. 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

4. 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)

5. 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 n sv su

6. 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,
In general,

7. First Fundamental Form
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

8. Curves on Surfaces Consider a curve c(t) = (U(t), V(t))
Velocity along the curve
So, I describes how much surface is “stretched”
Does not change if surface is “bent” without stretching

9. Areas on Surfaces For a region on a surface, area is given by
So, sqrt of det of matrix I gives stretch of area

10. Normal Curves
A normal curve is defined by the intersection of a surface and a plane containing the normal

11. 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

12. Finding Normal Curvatures
Consider a normal curve c(t) = (U(t), V(t)) n
n+dn
c(t) r r

13. 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

14. Second Fundamental Form
Note sign convention – this produces positive curvature for convex surfaces with outward-pointing normals

15. II and Second Derivatives
So II is related to the second derivatives of the surface

16. Normal Curvatures
The normal curvature in the direction w can be written as

17. 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

18. 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

19. Euler’s Formula Working in the Frénet frame,
Euler’s formula for normal curvature
 is the angle between w and d1

20. 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)

21. 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

22. Positive Gaussian Curvature: Elliptic Points
Convex/concave depending on sign of H
Tangent plane intersects surface at 1 point

23. Negative Gaussian Curvature: Hyperbolic Points
Tangent plane intersects surface along 2 curves

24. Zero Gaussian Curvature: Parabolic Points
Tangent plane intersects surface along 1 curve

25. 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/

26. 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:
Curvatures sensitive to noise (2nd derivative…), so sometimes just use sign of curvatures

27. 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.

28. 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