(Discrete) Differential Geometry
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1 (Discrete) Differential Geometry
2 Motivation Understand the structure of the surface Properties: smoothness, curviness, important directions How to modify the surface to change these properties What properties are preserved for different modifications The math behind the scenes for many geometry processing applications 2
3 Motivation Smoothness Mesh smoothing Curvature Adaptive simplification Parameterization 3
4 Motivation Triangle shape Remeshing Principal directions Quad remeshing 4
5 Differential Geometry MP M.P. docarmo: Differential Geometry of Curves and Surfaces, Prentice Hall, 1976 Leonard Euler ( ) Carl Friedrich Gauss ( ) 5
6 Parametric Curves velocity of particle on trajectory 6
7 Parametric Curves A Simple Example t a α 1 (t) = (a cos(t), a sin(t)) t [0,2π] 2 ] α 2 (t) = (a cos(2t), a sin(2t)) t [0,π] 7
8 Parametric Curves A Simple Example t α'(t) α(t) = (cos(t), sin(t)) α'(t) = (-sin(t) sin(t), cos(t)) α'(t) - direction of movement α'(t) - speed of movement 8
9 Parametric Curves A Simple Example α 2 '(t) t α 1 '(t) α 1 (t) = (cos(t), sin(t)) α 2 (t) = (cos(2t), sin(2t)) Same direction, different speed 9
10 Length of a Curve Let and 10
11 Length of a Curve Chord length Euclidean norm 11
12 Length of a Curve Chord length Arc length 12
13 Examples Straight line x(t) = (t,t), t [0, ) x(t) = (2t,2t), 2t) t [0, ) ) x(t) = (t 2,t 2 ), t [0, ) y(t) x(t) ( ) x(t) Circle x(t) = (cos(t),sin(t)),(t)), t [0,2π) ) y(t) x(t) x(t) 13
14 Examples α(t) = (a cos(t), a sin(t)), t [0,2π] α'(t) = (-a a sin(t), a cos(t)) Many possible parameterizations Length of the curve does not depend on parameterization! 14
15 Arc Length Parameterization Re-parameterization Arc length parameterization parameter value s for x(s) equals length of curve from x(a) to x(s) 15
16 Curvature x(t) a curve parameterized by arc length The curvature of x at t: the tangent vector at t the change in the tangent tvector attt R(t) = 1/κ (t) is the radius of curvature at t 16
17 Examples Straight line α(s) = us + v, u,v R 2 α'(s) = u α''(s) = 0 α''(s) = 0 Circle α(s) = (a cos(s/a), a sin(s/a)), s [0,2πa] α'(s) = (-sin(s/a), cos(s/a)) α''(s) = (-cos(s/a)/a, -sin(s/a)/a) α''(s) = 1/a 17
18 The Normal Vector x'(t) = T(t) - tangent vector x'(t) ( ) x'(t) - velocity x''(t) x''(t) = T'(t) - normal direction x''(t) -curvature x'(t) If x''(t) 0, define N(t) = T'(t)/ T'(t) Then x''(t) = T'(t) = κ(t)n(t) 18 x''(t)
19 The Osculating Plane The plane determined by the unit tangent and normal vectors T(s) and N(s) is called the osculating plane at s N N T T 19
20 The Binormal Vector For points s, s.t. κ(s) 0, the binormal vector B(s) is defined as: B(s) ) = T(s) ) N(s) ) T B B N N The binormal vector defines the osculating plane T 20
21 The Frenet Frame tangent normal binormal 21
22 The Frenet Frame {T(s), N(s), ) B(s)} form an orthonormal basis for R 3 called the Frenet frame How does the frame change when the particle il moves? What are T', N', B' in terms of T, N, B? 22
23 The Frenet Frame Frenet-Serret formulas curvature torsion (arc-length parameterization) 23
24 Curvature and Torsion Curvature: Deviation from straight line Torsion: Deviation from planarity Independent of parameterization intrinsic properties of the curve Invariant under rigid (translation+rotation) motion Define curve uniquely up to rigid motion 24
25 Curvature and Torsion Planes defined by x and two vectors: osculating plane: vectors T and N normal plane: vectors N and B rectifying plane: vectors T and B Osculating circle second order contact with curve center x radius C R 25
26 Differential Geometry: Surfaces 26
27 Differential Geometry: Surfaces Continuous surface Normal vector assuming regular parameterization, i.e. 27
28 Normal Curvature If x and x are orthogonal: u v 28
29 Normal Curvature 29
30 Surface Curvature Principal Curvatures maximum curvature minimum curvature Mean Curvature Gaussian Curvature 30
31 Principal Curvature Euler s Theorem: Planes of principal curvature are orthogonal and independent of parameterization. 31
32 Curvature 32
33 Surface Classification Isotropic k 1 = k 2 > 0 k 1 = k 2 = 0 Equal in all directions spherical planar Anisotropic Distinct principal directions k 1 > 0 k 2 > 0 k 1 > 0 k 2 =0 k 1 > 0 k 2 < 0 elliptic parabolic hyperbolic K > 0 K = 0 K < 0 developable 33
34 Principal Directions k2 k1 34
35 Gauss-Bonnet Theorem For ANY closed manifold surface with Euler number χ=2-2g: = 4 K( ) = K( ) K( ) = π 35
36 Gauss-Bonnet Theorem Example Sphere k 1 = k 2 = 1/r 1 2 K = k 1 k 2 = 1/r 2 Manipulate sphere New positive + negative curvature Cancel out! 36
37 Fundamental Forms First fundamental form Second fundamental form 37
38 Fundamental Forms I and II allow to measure length, angles, area, curvature arc element area element 38
39 Fundamental Forms Normal curvature = curvature of the normal curve at point Can be expressed in terms of fundamental forms as 39
40 Intrinsic Geometry Properties of the surface that only depend on the first fundamental form length angles Gaussian curvature (Theorema Egregium) 40
41 Laplace Operator Laplace operator gradient operator 2nd partial derivatives R scalar function in Euclidean space divergence operator Cartesian coordinates 41
42 Laplace-Beltrami Operator Extension to functions on manifolds Laplace- Beltrami gradient operator scalar function on manifold divergence operator 42
43 Laplace-Beltrami Operator For coordinate function(s) Laplace- Beltrami gradient operator mean curvature surface coordinate divergence normal function operator 43
44 Discrete Differential Operators Assumption: Meshes are piecewise linear approximations of smooth surfaces Approach: Approximate differential properties at point v as finite differences over local mesh neighborhood N(v) v = mesh vertex N d (v) = d-ring neighborhood Disclaimer: many possible discretizations, none is perfect v N 1 (v) 44
45 Functions on Meshes Function f givenatmeshverticesf(v vertices i ) = f(x i ) = f i Linear interpolation to triangle 1 B i (x) x i x k x j 45
46 Gradient of a Function 1 B i (x) Steepest ascent direction perpendicular to opposite edge x i x k x j Constant in the triangle 46
47 Gradient of a Function 47
48 Laplace operator: Discrete Laplace-Beltrami First Approach In 2D: On a grid finite it differences discretization: 48
49 Discrete Laplace-Beltrami Uniform Discretization v N 1 (v) Normalized: 49
50 Discrete Laplace-Beltrami Second dapproach Laplace-Beltrami operator: Compute integral around vertex Divergence theorem 50
51 Discrete Laplace-Beltrami Cotangent Formula Plugging in expression for gradients gives: 51
52 The Averaging g Region Edge midpoint θ Barycentric cell Voronoi cell Mixed cell c i = barycenter of triangle c i = circumcenter of triangle 52
53 The Averaging Region Mixed Cell If θ < π/2, c i is the circumcenter of the triangle (v i, v, v i+1 ) θ If θ π/2, c i is the midpoint of the edge (v i, v i+1 ) 53
54 Discrete Normal n ( x ) 54
55 Discrete Curvatures Mean curvature Gaussian curvature Principal curvatures 55
56 Example: Mean Curvature 56
57 Example: Gaussian Curvature original smoothed Discrete Gauss-Bonnet (Descartes) theorem: 57
58 References Discrete Differential-Geometry Operators for Triangulated 2- Manifolds, Meyer et al., 02 Restricted Delaunay triangulations and normal cycle, Cohen-Steiner et al., SoCG 03 On the convergence ence of metric and geometric properties of polyhedral surfaces, Hildebrandt et al., 06 Discrete Laplace operators: No free lunch, Wardetzkyet tal., SGP 07 58
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