Spring 2015 CSCI 599: Digital Geometry Processing 3.2 Classic Differential Geometry 2 Hao Li http://cs599.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 xu xv p x x n = u × v x x ⇥ u × v⇥ Assume regular parameterization x x = 0 normal exists u × v ⇥ 5 Angles on Surface Curve [ u ( t ) ,v ( t )] in uv-plane defines curve on the surface x(u, v) c(t)=x(u(t) ,v(t)) Two curves c 1 and c 2 intersecting at!p • angle of intersection? n • two tangents t and t 1 2 xu xv p ! ti = αixu + ⇥ixv c2 • compute inner product c1 tT t = cos θ t t 1 2 1 2 6 Angles on Surface Curve [ u ( t ) ,v ( t )] in uv-plane defines curve on the surface x(u, v) c(t)=x(u(t) ,v(t)) Two curves c 1 and c 2 intersecting at p T T t1 t2 =(α1xu + ⇥1xv) (α2xu + ⇥2xv) T T T = α1α2xu xu +(α1⇥2 + α2⇥1) xu xv + ⇥1⇥2xv xv T T xu xu xu xv α2 =(α1, ⇥1) T T xu xv xv xv ⇥2 ⇥ ⇥ 7 First Fundamental Form First fundamental form EF xT x xT x I = := u u u v FG xT x xT x ⇥ u v v v⇥ Defines inner product on tangent space T α α α α 1 , 2 := 1 I 2 ⇥ ⇥ ⇥ ⇥ ⇤ 1⇥ 2⇥⌅ 1⇥ 2⇥ 8 First Fundamental Form First fundamental form I allows to measure! (w.r.t. surface metric) Angles t>t = (↵ , β ), (↵ , β ) 1 2 h 1 1 2 2 i Length ds2 = (du, dv), (du, dv) squared ! ⇥ infinitesimal ! = Edu2 +2F dudv + Gdv2 length Area dA = x x du dv ⌅ u ⇤ v⌅ T T T 2 = x xu x xv (x xv) du dv infinitesimal ! u · v − u Area ⇥ = EG F 2du dv cross product →− determinant with unit vectors → area 9 Sphere Example Spherical parameterization cos u sin v x(u, v)= sin u sin v , (u, v) [0, 2π) [0, π) cos v ⇥ ⇥ × Tangent vectors⇤ ⌅ sin u sin v cos u cos v − x (u, v)= cos u sin v x (u, v)= sin u cos v u v ⇥ 0 ⇥ sin v − ⇤ ⌅ ⇤ ⌅ First fundamental Form sin2 v 0 I = 01 ⇥ 10 Sphere Example Length of equator x(t, ⇡/2) 2π 2π 2 2 1ds = E (ut) +2Futvt + G (vt) dt 0 0 ⇥ 2π = sin v dt 0 =2π sin v =2π 11 Sphere Example Area of a sphere π 2π π 2π 1dA = EG F 2 du dv 0 0 0 0 − ⇥ π 2π = sin v du dv 0 0 =4π 12 Normal Curvature Tangent vector t … n xu xv p t xu xv t = cos φ + sin φ ∥xu∥ ∥xv∥ unit vector 13 Normal Curvature … defines intersection plane, yielding curve c(t) normal curve n t c(t) p xu xv t = cos φ + sin φ ∥xu∥ ∥xv∥ 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 Normal curvature n ( t ) is defined as curvature of the normal curve c ( t ) at point p(t)=x(u, v) With second fundamental form ef xT nxT n II = := uu uv fg xT nxT n ⇥ ⇤ uv vv ⌅ normal curvature can be computed as T 2 2 ¯t II¯t ea +2fab + gb t = axu + bxv κ (¯t)= = n ¯tT I ¯t Ea2 +2F ab + Gb2 ¯t =(a, b) 16 Surface Curvature(s) Principal curvatures! • Maximum curvature κ1 = max κn(φ) φ • Minimum curvature κ2 = min κn(φ) φ 2 2 • Euler theorem κn(⇥)=κ1 cos ⇥ + κ2 sin ⇥ • Corresponding principal directions e 1 , e 2 are orthogonal 17 Surface Curvature(s) Principal curvatures! • Maximum curvature κ1 = max κn(φ) φ • Minimum curvature κ2 = min κn(φ) φ 2 2 • Euler theorem κn(⇥)=κ1 cos ⇥ + κ2 sin ⇥ • Corresponding principal directions e 1 , e 2 are orthogonal Special curvatures! κ1 + κ2 • Mean curvature H = extrinsic 2 • Gaussian curvature K = κ1 · κ2 intrinsic (only first FF) 18 Invariants Gaussian and mean curvature! • determinant and trace only ! ! ! • eigenvalues and orthovectors 19 Mean Curvature Integral representations 20 Curvature of Surfaces κ + κ Mean curvature! H = 1 2 2 • H =0 everywhere minimal surface ! soap film 21 Curvature of Surfaces κ + κ Mean curvature! H = 1 2 2 • H =0 everywhere minimal surface ! Green Void, Sydney Architects: Lava 22 Curvature of Surfaces Gaussian curvature! K = κ1 · κ2 • K =0 everywhere developable surface ! 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) 6πr − 3C(r) K = lim r→0 πr3 Gaussian curvature of a surface is invariant under local isometry 25 Classification Point x on the surface is called! • elliptic, if K>0 • hyperbolic, if K<0 Gaussian curvature K • parabolic, if K =0 • umbilic, if κ1 = κ2 or isotropic 26 Classification Point x on the surface is called 27 Gauss-Bonnet Theorem For any closed manifold surface with Euler characteristic χ =2 2g − K =2⇥ Z K()=K()=K()=4π Z Z Z 28 Gauss-Bonnet Theorem Sphere κ1 = κ2 =1/r 2 K = κ1κ2 =1/r 1 K =4πr2 =4π · r2 Z when sphere is deformed, new positive and negative curvature cancel out 29 Differential Operators Gradient! @f @f ! f := ,..., r @x @x ✓ 1 n ◆ ! • points in the direction of the steepest ascend 30 Differential Operators Divergence! ! @F @F divF = F := 1 + ...+ n ! r· @x1 @xn • 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 @F @F divF = F := 1 + ...+ n r· @x1 @xn high divergence low divergence 32 Laplace Operator gradient 2nd partial Laplace operator derivatives operator ∂2f ∆ = div∇ = f f ! 2 i ∂xi Cartesian coordinates function in divergence Euclidean space operator 33 Laplace-Beltrami Operator Extension of Laplace fo functions on manifolds Laplace- gradient Beltrami operator …of the surface ∆S f = divS ∇S f function on divergence manifold S operator Laplace on the surface 34 Laplace-Beltrami Operator Laplace- gradient Beltrami operator mean curvature ∆S x = divS ∇S x = −2Hn surface function on divergence normal manifold S 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://cs599.hao-li.com Thanks! 38.