1. 2D/3D Shape Manipulation, 3D Printing
CS 6501
2D/3D Shape Manipulation, 3D Printing
Discrete Differential Geometry
Surfaces
Slides from Olga Sorkine, Eitan Grinspun

2. Surfaces, Parametric Form
Continuous surface
Tangent plane at point p(u,v) is spanned by n pu pv
p(u,v) v u

3. Isoparametric Lines
Lines on the surface when keeping one parameter fixed v u

4. Surface Normals Surface normal: n pv pu
Assuming regular parameterization, i.e., n pu pv
p(u,v) v u

5. Normal Curvature n pv pu p t
Direction t in the tangent plane (if pu and pv are orthogonal): t 
Tangent plane

6. Normal Curvature n() = ((p)) n pv pu p t 
The curve  is the intersection of the surface with the plane through n and t.
Normal curvature:
n() = ((p))
Make video of the normal plane? t 
Tangent plane

7. Surface Curvatures Principal curvatures Mean curvature
Maximal curvature
Minimal curvature
Mean curvature
Gaussian curvature

8. Principal Directions
Principal directions: tangent vectors corresponding to max and min t2 t1
 max
min curvature
max curvature
tangent plane

9. Principal Directions
תמונה
Euler’s Theorem: Planes of principal curvature are orthogonal
and independent of parameterization.

10. Principal Directions
תמונה

11. Mean Curvature
Intuition for mean curvature

12. Classification A point p on the surface is called
Elliptic, if K > 0
Parabolic, if K = 0
Hyperbolic, if K < 0
Umbilical, if
Developable surface iff K = 0

13. Local Surface Shape By Curvatures
K > 0, 1= 2
K = 0
Isotropic: all directions are principal directions
spherical (umbilical)
planar
2 > 0, 1 > 0
2 = 0
1 > 0
2 < 0
K > 0
K = 0
K < 0
Anisotropic: 2 distinct principal directions
elliptic
parabolic
hyperbolic

14. Gauss-Bonnet Theorem
For a closed surface M:

15. Gauss-Bonnet Theorem For a closed surface M:
Compare with planar curves:

16. Fundamental Forms First fundamental form Second fundamental form
Together, they define a surface (given some compatibility conditions)
Intuitive explanation for I and II!

17. Fundamental Forms I and II allow to measure
length, angles, area, curvature
arc element
area element
here is just I, maybe add things that II measures…

18. Intrinsic Geometry
Properties of the surface that only depend on the first fundamental form
length
angles
Gaussian curvature (Theorema Egregium)

19. Laplace Operator gradient operator 2nd partial derivatives
function in Euclidean space
Cartesian coordinates
divergence operator

20. Laplace-Beltrami Operator
Extension of Laplace to functions on manifolds
gradient operator
Laplace- Beltrami
function on surface M
divergence operator

21. Laplace-Beltrami Operator
For coordinate functions:
gradient operator
mean curvature
Laplace- Beltrami
unit surface normal
function on surface M
divergence operator

22. Differential Geometry on Meshes
Assumption: meshes are piecewise linear approximations of smooth surfaces
Can try fitting a smooth surface locally (say, a polynomial) and find differential quantities analytically
But: it is often too slow for interactive setting and error prone

23. Discrete Differential Operators
Approach: approximate differential properties at point v as spatial average over local mesh neighborhood N(v) where typically
v = mesh vertex
Nk(v) = k-ring neighborhood

24. Discrete Laplace-Beltrami
Uniform discretization: L(v) or ∆v
Depends only on connectivity = simple and efficient
Bad approximation for irregular triangulations vi vj

25. Discrete Laplace-Beltrami
Intuition for uniform discretization

26. Discrete Laplace-Beltrami
Intuition for uniform discretization
vi-1 vi
vi+1 

27. Discrete Laplace-Beltrami
Intuition for uniform discretization
vj1
vj2 vi
vj6
vj3
vj5
vj4

28. Discrete Laplace-Beltrami
Cotangent formula Ai vi vj
ij
ij

29. Voronoi Vertex Area
Unfold the triangle flap onto the plane (without distortion) vi θ vj

30. Voronoi Vertex Area vi vj needs a better picture and definition! θ
Flattened flap vi vi θ
cj+1 cj vj
needs a better picture and definition!

31. Discrete Laplace-Beltrami
Cotangent formula
Accounts for mesh geometry
Potentially negative/ infinite weights

32. Discrete Laplace-Beltrami
Cotangent formula
Can be derived using linear Finite Elements
Nice property: gives zero for planar 1-rings!

33. Discrete Laplace-Beltramivi
Uniform Laplacian Lu(vi)
Cotangent Laplacian Lc(vi)
Mean curvature normal b a vj

34. Discrete Laplace-Beltramivi
Uniform Laplacian Lu(vi)
Cotangent Laplacian Lc(vi)
Mean curvature normal
For nearly equal edge lengths Uniform ≈ Cotangent b a vj

35. Discrete Laplace-Beltramivi
Uniform Laplacian Lu(vi)
Cotangent Laplacian Lc(vi)
Mean curvature normal
For nearly equal edge lengths Uniform ≈ Cotangent b a vj
Cotan Laplacian allows computing discrete normal

36. Discrete Curvatures Mean curvature (sign defined according to normal)
Gaussian curvature
Principal curvatures Ai j

37. Discrete Gauss-Bonnet Theorem
Total Gaussian curvature is fixed for a given topology

38. Example: Discrete Mean Curvature

39. Links and Literature
M. Meyer, M. Desbrun, P. Schroeder, A. Barr Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, VisMath, 2002

40. Links and Literature
P. Alliez, Estimating Curvature Tensors on Triangle Meshes, Source Code
principal directions

41. Measuring Surface Smoothness

42. Links and Literature
Grinspun et al.:Computing discrete shape operators on general meshes, Eurographics 2006

43. Reflection Lines as an Inspection Tool
Shape optimization using reflection lines E. Tosun, Y. I. Gingold, J. Reisman, D. Zorin Symposium on Geometry Processing 2007

44. Reflection Lines as an Inspection Tool
Shape optimization using reflection lines E. Tosun, Y. I. Gingold, J. Reisman, D. Zorin Symposium on Geometry Processing 2007

45. Thank You