Spring 2018 CSCI 621: Digital Geometry Processing 4.1 Classic Differential Geometry 2 Hao Li http://cs621.hao-li.com 1
Outline • Parametric Curves • Parametric Surfaces 2
Surfaces What characterizes shape? • shape does not depend on Euclidean motions • metric and curvatures • smooth continuous notions to discrete notions 3
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 4
Parametric Surfaces Continuous surface x ( u, v ) x ( u, v ) = y ( u, v ) z ( u, v ) n Normal vector x u x v p x u � x v n = ⇥ x u � x v ⇥ Assume regular parameterization x u � x v ⇥ = 0 normal exists 5
Angles on Surface [ u ( t ) , v ( t )] Curve in uv-plane defines curve on the x ( u, v ) surface c ( t ) = x ( u ( t ) , v ( t )) Two curves and intersecting at c 1 c 2 p • angle of intersection? n • two tangents and t 1 t 2 x u x v t i = α i x u + β i x v p c 2 • compute inner product c 1 t T 1 t 2 = cos θ � t 1 � � t 2 � 6
Angles on Surface [ u ( t ) , v ( t )] Curve in uv-plane defines curve on the x ( u, v ) surface c ( t ) = x ( u ( t ) , v ( t )) Two curves and intersecting at c 1 c 2 p 1 t 2 = ( α 1 x u + β 1 x v ) T ( α 2 x u + β 2 x v ) t T = α 1 α 2 x T u x u + ( α 1 β 2 + α 2 β 1 ) x T u x v + β 1 β 2 x T v x v � x T x T ⇥ � α 2 ⇥ u x u u x v = ( α 1 , β 1 ) x T x T β 2 u x v v x v 7
First Fundamental Form First fundamental form � E ⇥ � x T x T ⇥ F u x u u x v I = := x T x T F G u x v v x v Defines inner product on tangent space ⇥ T ⇤� α 1 ⇥ � α 2 ⇥⌅ � α 1 � α 2 ⇥ I := β 1 β 2 β 1 β 2 , 8
First Fundamental Form First fundamental form allows to measure I (w.r.t. surface metric) t > 1 t 2 = h ( α 1 , β 1 ) , ( α 2 , β 2 ) i Angles d s 2 = � (d u, d v ) , (d u, d v ) ⇥ squared Length infinitesimal E d u 2 + 2 F d u d v + G d v 2 = length d A = ⌅ x u ⇤ x v ⌅ d u d v Area ⇥ u x v ) 2 d u d v x T u x u · x T v x v � ( x T = infinitesimal Area � = EG � F 2 d u d v cross product → determinant with unit vectors → area 9
Sphere Example Spherical parameterization cos u sin v sin u sin v x ( u, v ) = ( u, v ) ∈ [0 , 2 π ) × [0 , π ) , cos v Tangent vectors − sin u sin v cos u cos v cos u sin v x v ( u, v ) = sin u cos v x u ( u, v ) = − sin v 0 First fundamental Form sin 2 v � ⇥ 0 I = 0 1 10
Sphere Example Length of equator x ( t, π / 2) � 2 π � 2 π ⇥ E ( u t ) 2 + 2 Fu t v t + G ( v t ) 2 d t 1 d s = 0 0 � 2 π = sin v d t 0 = 2 π sin v = 2 π 11
Sphere Example Area of a sphere � 2 π � 2 π � π � π ⇥ EG − F 2 d u d v 1 d A = 0 0 0 0 � 2 π � π = sin v d u d v 0 0 = 4 π 12
Normal Curvature Tangent vector … t n x u x v p t t = cos φ x u ∥ x u ∥ + sin φ x v ∥ x v ∥ unit vector 13
Normal Curvature … defines intersection plane, yielding curve c ( t ) normal curve n c ( t ) t p t = cos φ x u ∥ x u ∥ + sin φ x v ∥ x v ∥ 14
Geometry of the Normal Gauss map • normal at point • consider curve in surface again • study its curvature at p • normal “tilts” along curve 15
Normal Curvature κ n ( t ) Normal curvature is defined as curvature of the c ( t ) p ( t ) = x ( u, v ) normal curve at point With second fundamental form ⇤ ⌅ x T x T � ⇥ uu n uv n e f II = := f g x T x T uv n vv n normal curvature can be computed as ea 2 + 2 fab + gb 2 t T II ¯ ¯ t = a x u + b x v t κ n (¯ t ) = = Ea 2 + 2 Fab + Gb 2 t T I ¯ ¯ ¯ = ( a, b ) t t 16
Surface Curvature(s) Principal curvatures κ 1 = max κ n ( φ ) • Maximum curvature φ κ 2 = min φ κ n ( φ ) • Minimum curvature κ n ( φ ) = κ 1 cos 2 φ + κ 2 sin 2 φ • Euler theorem • Corresponding principal directions , are orthogonal e 1 e 2 17
Surface Curvature(s) Principal curvatures κ 1 = max κ n ( φ ) • Maximum curvature φ κ 2 = min φ κ n ( φ ) • Minimum curvature κ n ( φ ) = κ 1 cos 2 φ + κ 2 sin 2 φ • Euler theorem • Corresponding principal directions , are orthogonal e 1 e 2 Special curvatures H = κ 1 + κ 2 • Mean curvature extrinsic 2 • Gaussian curvature K = κ 1 · κ 2 intrinsic (only first FF) 18
Invariants Gaussian and mean curvature • determinant and trace only • eigenvalues and orthovectors 19
Mean and Gaussian Curvature Integral representations 20
Curvature of Surfaces H = κ 1 + κ 2 Mean curvature 2 • everywhere minimal surface H = 0 → soap film 21
Curvature of Surfaces H = κ 1 + κ 2 Mean curvature 2 • everywhere minimal surface H = 0 → Green Void, Sydney Architects: Lava 22
Curvature of Surfaces Gaussian curvature K = κ 1 · κ 2 • everywhere developable surface K = 0 → surface that can be flattened to a plane without distortion (stretching or compression) Disney, Concert Hall, L.A. Timber Fabric Architects: Gehry Partners IBOIS, EPFL 23
Shape Operator Derivative of Gauss map • second fundamental form • local coordinates 24
Intrinsic Geometry Properties of the surface that only depend on the first fundamental form • length • angles • Gaussian curvature (Theorema Egregium) remarkable theorem (Gauss) 6 π r − 3 C ( r ) K = lim π r 3 r → 0 Gaussian curvature of a surface is invariant under local isometry 25
Classification Point on the surface is called x • elliptic, if K > 0 • hyperbolic, if K < 0 Gaussian curvature K • parabolic, if K = 0 • umbilic, if κ 1 = κ 2 or isotropic 26
Classification Point on the surface is called x 27
Gauss-Bonnet Theorem For any closed manifold surface with Euler χ = 2 − 2 g characteristic Z K = 2 πχ Z Z Z K ( ) = K ( ) = K ( ) = 4 π 28
Gauss-Bonnet Theorem Sphere κ 1 = κ 2 = 1 /r K = κ 1 κ 2 = 1 /r 2 Z K = 4 π r 2 · 1 r 2 = 4 π when sphere is deformed, new positive and negative curvature cancel out 29
Differential Operators Gradient ✓ ∂ f ◆ , . . . , ∂ f r f := ∂ x 1 ∂ x n • points in the direction of the steepest ascend 30
Differential Operators Divergence div F = r · F := ∂ F 1 + . . . + ∂ F n ∂ x 1 ∂ x 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 31
Differential Operators Divergence div F = r · F := ∂ F 1 + . . . + ∂ F n ∂ x 1 ∂ x n high divergence low divergence 32
Laplace Operator gradient 2nd partial Laplace operator derivatives operator ∂ 2 f � ∆ f = div ∇ f = ∂ x 2 i i Cartesian coordinates function in divergence Euclidean space operator 33
Laplace-Beltrami Operator Extension of Laplace of functions on manifolds Laplace- gradient Beltrami operator …of the surface ∆ S f = div S ∇ S f function on divergence S manifold operator Laplace on the surface 34
Laplace-Beltrami Operator Laplace- gradient Beltrami operator mean curvature ∆ S x = div S ∇ S x = − 2 H n surface function on normal divergence S manifold operator 35
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 • http://graphics.stanford.edu/courses/cs468-13-spring/schedule.html 36
Next Time Discrete Differential Geometry 37
http://cs621.hao-li.com Thanks! 38
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