3.1 Classic Differential Geometry

Transcription

1 Spring 2015 CSCI 599: Digital Geometry Processing 3.1 Classic Differential Geometry Hao Li 1

2 Spring 2014 CSCI 599: Digital Geometry Processing 3.1 Classic Differential Geometry With a Twist! Hao Li 2

3 Administrative Exercise handouts: 11:59 PM on Monday Office hours later from 2pm to 3pm 3

4 Some Updates: run.usc.edu/vega Another awesome free library with half-edge data-structure By Prof. Jernej Barbic 4

5 FYI MeshLab! Popular Mesh Processing Software (meshlab.sourceforge.net) 5

6 FYI BeNTO3D! Mesh Processing Framework for Mac ( 6

7 Last Time Discrete Representations! Explicit (parametric, polygonal meshes) Implicit Surfaces (SDF, grid representation) Geometry Topology Conversions E I: Closest Point, SDF, Fast Marching I E: Marching Cubes Algorithm 7

8 Differential Geometry Why do we care?! Geometry of surfaces Mothertongue of physical theories Computation: processing / simulation 8

9 Motivation We need differential geometry to compute! surface curvature paramaterization distortion deformation energies 9

10 Applications: 3D Reconstruction 10

11 Applications: Head Modeling 11

12 Applications: Facial Animation 12

13 Motivation Geometry is the key! studied for centuries (Cartan, Poincaré, Lie, Hodge, de Rham, Gauss, Noether ) mostly differential geometry differential and integral calculus invariants and symmetries 13

14 Getting Started How to apply DiffGeo ideas?! surfaces as a collection of samples and topology (connectivity) apply continuous ideas BUT: setting is discrete what is the right way? discrete vs. discretized Let s look at that first 14

15 Getting Started What characterizes structure(s)?! What is shape? Euclidean Invariance What is physics? Conservation/Balance Laws What can we measure? area, curvature, mass, flux, circulation 15

16 Getting Started Invariant descriptors! quantities invariant under a set of transformations Intrinsic descriptor! quantities which do not depend on a coordinate frame 16

17 Outline Parametric Curves Parametric Surfaces Formalism & Intuition 17

18 Differential Geometry Leonard Euler ( ) Carl Friedrich Gauss ( ) 18

19 Parametric Curves x :[a, b] IR IR 3 x(b) a t b x(t) xt(t) x(a) x(t) = x(t) y(t) z(t) x t (t) := dx(t) dt = dx(t) /dt dy(t) /dt dz(t) /dt 19

20 Recall: Mappings Injective NO SELF-INTERSECTIONS! Surjective SELF-INTERSECTIONS! Bijective AMBIGUOUS PARAMETERIZATION 20

21 Parametric Curves A parametric curve x(t) is! x(t) simple: is injective (no self-intersections) differentiable: x t (t) is defined for all regular: for all t 2 [a, b] x t (t) 6= 0 t 2 [a, b] x(b) x(t) xt(t) x(a) 21

22 Length of a Curve Let t i = a + i t and x i = x(t i ) x i x(a) x(b) a t t i b 22

23 Length of a Curve Polyline chord length S = i x i = i x i t t, x i := x i+1 x i norm change Curve arc length ( t! 0 ) x i t s = s(t) = a x t dt x(a) x(b) length =! integration of infinitesimal change! norm of speed a t t i b 23

24 Re-Parameterization Mapping of parameter domain u :[a, b] [c, d] Re-parameterization w.r.t. u(t) [c, d] IR 3, t x(u(t)) Derivative (chain rule) dx(u(t)) dt = dx du du dt = x u (u(t)) u t (t) 24

25 Re-Parameterization Example f : 0, 1 2 IR 2, t (4t, 2t) : 0, 1 2 [0, 1], t 2t g : [0, 1] IR 2, t (2t, t) g( (t)) = f(t) 25

26 Arc Length Parameterization Mapping of parameter domain: t s(t) = a t x t dt Parameter s for x(s) equals length from x(a) to x(s) x(s) = x(s(t)) ds = x t dt Special properties of resulting curve same infinitesimal change x s (s) =1, x s (s) x ss (s) =0 defines orthonormal frame 26

27 The Frenet Frame Taylor expansion x(t + h) = x(t) +x t (t) h x tt(t) h x ttt(t) h for convergence analysis and approximations Define local frame (t, n, b) (Frenet frame) t = x t b = x t x tt n = b t x t x t x tt tangent main normal binormal 27

28 The Frenet Frame Orthonormalization of local frame x ttt x tt n b x t t local affine frame Frenet frame 28

29 The Frenet Frame Frenet-Serret: Derivatives w.r.t. arc length s t s = + n n s = t + b b s = n Curvature (deviation from straight line) = x ss n b Torsion (deviation from planarity) t = 1 2 det([x s, x ss, x sss ]) 29

30 Curvature and Torsion Planes defined by x and two vectors:! osculating plane: vectors and normal plane: vectors n and rectifying plane: vectors t and Osculating circle! second order contact with curve t b b n n b t center radius c = x +(1/apple)n 1/apple 30

31 Curvature and Torsion Curvature: Deviation from straight line Torsion: Deviation from planarity Independent of parameterization intrinsic properties of the curve Euclidean invariants invariant under rigid motion Define curve uniquely up to a rigid motion 31

32 Curvature: Some Intuition A line through two points on the curve (Secant) 32

33 Curvature: Some Intuition A line through two points on the curve (Secant) 33

34 Curvature: Some Intuition Tangent, the first approximation limiting secant as the two points come together 34

35 Curvature: Some Intuition Circle of curvature Consider the circle passing through 3 pints of the curve 35

36 Curvature: Some Intuition Circle of curvature The limiting circle as three points come together 36

37 Curvature: Some Intuition Radius of curvature r 37

38 Curvature: Some Intuition Radius of curvature r 38

39 Curvature: Some Intuition Signed curvature Sense of traversal along curve 39

40 Curvature: Some Intuition Gauß map Point on curve maps to point on unit circle 40

41 Curvature: Some Intuition Shape operator (Weingarten map) Change in normal as we slide along curve negative directional derivative D of Gauß map describes directional curvature using normals as degrees of freedom! accuracy/convergence/implementation (discretization) 41

42 Curvature: Some Intuition Turning number, k Number of orbits in Gaussian image 42

43 Curvature: Some Intuition Turning number theorem For a closed curve, the integral of curvature is an integer multiple of 2π 43

44 Take Home Message In the limit of a refinement sequence, discrete measure of length and curvature agree with continuous measures 44

45 Outline Parametric Curves Parametric Surfaces 45

46 Surfaces What characterizes shape?! shape does not depend on Euclidean motions metric and curvatures smooth continuous notions to discrete notions 46

47 Metric on Surfaces Measure Stuff! angle, length, area requires an inner product we have: Euclidean inner product in domain we want to turn this into: inner product on surface 47

48 Parametric Surfaces Continuous surface x(u, v) = Normal vector x(u, v) y(u, v) z(u, v) n = x u x v x u x v x u n p x v Assume regular parameterization x u x v = 0 normal exists 48

49 Angles on Surface Curve [u(t),v(t)] surface x(u, v) in uv-plane defines curve on the c(t) = x(u(t),v(t)) Two curves c 1 and c 2 intersecting at! p angle of intersection? t 1 t 2 two tangents and x u n x v! t i = i x u + i x v p c 2 compute inner product c 1 t T 1 t 2 = cos t 1 t 2 49

50 Angles on Surface Curve [u(t),v(t)] surface x(u, v) in uv-plane defines curve on the c(t) = x(u(t),v(t)) Two curves c 1 and c 2 intersecting at p t T 1 t 2 = ( 1 x u + 1 x v ) T ( 2 x u + 2 x v ) = 1 2 x T u x u +( ) x T u x v x T v x v = ( 1, 1 ) x T u x u x T u x v x T u x v x T v x v

51 First Fundamental Form First fundamental form I = E F F G := xt u x u x T u x v x T u x v x T v x v Defines inner product on tangent space 1 1, 2 2 := 1 1 T I

52 First Fundamental Form First fundamental form (w.r.t. surface metric) I allows to measure! Angles t > 1 t 2 = h( 1, 1), ( 2, 2)i Length ds 2 = (du, dv), (du, dv) = Edu 2 +2F dudv + Gdv 2 squared! infinitesimal! length Area infinitesimal! Area da = x u x v du dv = x T u x u x T v x v (x T u x v ) 2 du dv = EG F 2 du dv cross product determinant with unit vectors area 52

53 Sphere Example Spherical parameterization x(u, v) = cos u sin v sin u sin v cos v, (u, v) [0, 2 ) [0, ) Tangent vectors x u (u, v) = sin u sin v cos u sin v 0 x v (u, v) = cos u cos v sin u cos v sin v First fundamental Form I = sin 2 v

54 Sphere Example Length of equator x(t, /2) 0 2 1ds = 0 2 E (u t ) 2 +2Fu t v t + G (v t ) 2 dt = 0 2 sin v dt = 2 sin v = 2 54

55 Sphere Example Area of a sphere dA = EG F 2 du dv = sin v du dv = 4 55

56 Normal Curvature Tangent vector t n x u x v t p t = cos φ x u x u + sin φ x v x v unit vector 56

57 Normal Curvature defines intersection plane, yielding curve c(t) normal curve n t p c(t) t = cos φ x u x u + sin φ x v x v 57

58 Geometry of the Normal Gauss map!!!! normal at point consider curve in surface again study its curvature at p normal tilts along curve 58

59 Normal Curvature Normal curvature apple n (t) is defined as curvature of the normal curve c(t) at point p(t) =x(u, v) With second fundamental form II = e f x T uu n := f g x T uvn x T uvn x T vvn normal curvature can be computed as n( t) = t T II t t T I t = ea2 +2fab + gb 2 t = ax u + bx v Ea 2 +2Fab + Gb 2 t = (a, b) 59

60 Surface Curvature(s) Principal curvatures! Maximum curvature Minimum curvature Euler theorem κ 1 = max Corresponding principal directions, are orthogonal φ κ n(φ) κ 2 = min κ n (φ) φ n( ) = 1 cos sin 2 e 1 e 2 60

61 Surface Curvature(s) Principal curvatures! Maximum curvature Minimum curvature Euler theorem κ 1 = max Corresponding principal directions, are orthogonal φ κ n(φ) κ 2 = min κ n (φ) φ n( ) = 1 cos sin 2 e 1 e 2 Special curvatures! Mean curvature Gaussian curvature H = K = κ 1 κ 2 extrinsic intrinsic (only first FF) 61

62 Invariants Gaussian and mean curvature!!!! determinant and trace only eigenvalues and orthovectors 62

63 Mean Curvature Integral representations 63

64 Curvature of Surfaces Mean curvature! H = H =0 everywhere! minimal surface soap film 64

65 Curvature of Surfaces Mean curvature! H = H =0 everywhere! minimal surface Green Void, Sydney Architects: Lava 65

66 Curvature of Surfaces Gaussian curvature! K = κ 1 κ 2 K =0everywhere! developable surface surface that can be flattened to a plane without distortion (stretching or compression) Disney, Concert Hall, L.A. Architects: Gehry Partners Timber Fabric IBOIS, EPFL 66

67 Shape Operator Derivative of Gauss map! second fundamental form!!! local coordinates 67

68 Intrinsic Geometry Properties of the surface that only depend on the! first fundamental form! length angles Gaussian curvature (Theorema Egregium) remarkable theorem (Gauss) K = lim r 0 6πr 3C(r) πr 3 Gaussian curvature of a surface is invariant under local isometry 68

69 Classification Point x on the surface is called! elliptic, if K>0 hyperbolic, if parabolic, if K<0 K =0 Gaussian curvature K umbilic, if κ 1 = κ 2 or isotropic 69

70 Classification Point x on the surface is called 70

71 Gauss-Bonnet Theorem For any closed manifold surface with Euler characteristic =2 2g Z K =2 Z K( ) = Z K( ) = Z K( ) = 4 71

72 Gauss-Bonnet Theorem Sphere 1 = 2 =1/r K = 1 2 =1/r 2 Z K =4 r 2 1 r 2 =4 when sphere is deformed, new positive and negative curvature cancel out 72

73 Differential Operators Gradient!! @x n! points in the direction of the steepest ascend 73

74 Differential Operators Divergence!!! divf = r F 1 n volume density of outward flux of vector field magnitude of source or sink at given point Example: incompressible fluid velocity field is divergence-free 74

75 Differential Operators Divergence divf = r F 1 n high divergence low divergence 75

76 Laplace Operator Laplace operator gradient operator 2nd partial derivatives f = div f = i x 2 i 2 f function in Euclidean space divergence operator Cartesian coordinates 76

77 Laplace-Beltrami Operator Extension of Laplace fo functions on manifolds Laplace- Beltrami gradient operator of the surface S f = div S S f function on manifold S divergence operator Laplace on the surface 77

78 Laplace-Beltrami Operator Laplace- Beltrami gradient operator mean curvature S x = div S S x = 2Hn function on manifold S divergence operator surface normal 78

79 Literature M. Do Carmo: Differential Geometry of Curves and Surfaces, Prentice Hall, 1976 A. Pressley: Elementary Differential Geometry, Springer, 2010 G. Farin: Curves and Surfaces for CAGD, Morgan Kaufmann, 2001 W. Boehm, H. Prautzsch: Geometric Concepts for Geometric Design, AK Peters 1994 H. Prautzsch, W. Boehm, M. Paluszny: Bézier and B-Spline Techniques, Springer 2002 ddg.cs.columbia.edu 79

80 Next Time Discrete Differential Geometry 80

81 Thanks! 81