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Differential Geometry Spring 2014 CSCI 599: Digital Geometry Processing 3.1 Classic Differential Geometry Hao Li http://cs599.hao-li.com !1 Spring 2014 CSCI 599: Digital Geometry Processing 3.1 Classic Differential Geometry With a Twist! Hao Li http://cs599.hao-li.com !2 Administrative • Exercise handouts: 11:59 PM every 2nd Wednesday • My first office hours later from 2pm to 4pm • This week only lecture. !3 Some Updates: run.usc.edu/vega Another awesome free library with half-edge data-structure By Prof. Jernej Barbic !4 FYI MeshLab! Popular Mesh Processing Software (meshlab.sourceforge.net) !5 FYI BeNTO3D! Mesh Processing Framework for Mac (www.bento3d.com) !6 Last Time Discrete Representations! • Explicit (parametric, polygonal meshes) Geometry • Implicit Surfaces (SDF, grid representation) Topology • Conversions • E→I: Closest Point, SDF, Fast Marching • I→E: Marching Cubes Algorithm !7 Differential Geometry Why do we care?! • Geometry of surfaces • Mothertongue of physical theories • Computation: processing / simulation !8 Motivation We need differential geometry to compute! • surface curvature • paramaterization distortion • deformation energies !9 Applications: 3D Reconstruction !10 Applications: Head Modeling !11 Applications: Facial Animation !12 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 Getting Started How to apply DiffGeo ideas?! • surfaces as a collecition 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 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 Getting Started Invariant descriptors! • quantities invariant under a set of transformations Intrinsic descriptor! • quantities which do notd depend on a coordinate frame !16 Outline • Parametric Curves • Parametric Surfaces Formalism & Intuition !17 Differential Geometry Leonard Euler (1707-1783) Carl Friedrich Gauss (1777-1855) !18 Parametric Curves x :[a, b] IR IR 3 ⊂ ⇥ x(b) x(t) a t b xt(t) x(a) x(t) dx(t) /dt dx(t) x(t)= y(t) x (t) := = dy(t) /dt t dt z(t) ⇥ dz(t) /dt ⇥ ⇤ ⌅ ⇤ ⌅ !19 Recall: Mappings Injective Surjective Bijective NO SELF-INTERSECTIONS! SELF-INTERSECTIONS! AMBIGUOUS PARAMETERIZATION !20 Parametric Curves A parametric curve x ( t ) is! • simple: x ( t ) is injective (no self-intersections) • differentiable: x ( t ) is defined for all t [a, b] t 2 • regular: x ( t ) =0 for all t [a, b] t 6 2 x(b) x(t) xt(t) x(a) !21 Length of a Curve Let t i = a + i ∆ t and xi = x(ti) xi x(a) x(b) a ∆t ti b !22 Length of a Curve Polyline chord length ∆x S = ∆x = i ∆t, ∆x := x x ⇥ i⇥ ∆t i ⇥ i+1 − i⇥ i i norm change ⇥ ⇥ Curve arc length ( ∆ t 0 ) x ! i t s = s(t)= xt dt a x(a) x(b) length = ! integration of infinitesimal change ! a ∆t ti b " norm of speed !23 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)) dx du = = x (u(t)) u (t) dt du dt u t !24 Re-Parameterization Example 1 f : 0, IR 2 ,t (4t, 2t) 2 → ⇥ ⇥ 1 φ : 0, [0, 1] ,t 2t 2 → ⇥ ⇥ g : [0, 1] IR 2 ,t (2t, t) → ⇥ g(φ(t)) = f(t) ⇒ !25 Arc Length Parameterization Mapping of parameter domain: t t s(t)= x dt ⇥ ⇤ t⇤ a Parameter s for x ( s ) equals length from x ( a ) to x(s) x(s)=x(s(t)) ds = x dt t same infinitesimal change Special properties of resulting curve x (s) =1, x (s) x (s)=0 ⇥ s ⇥ s · ss defines orthonormal frame !26 The Frenet Frame Taylor expansion 1 1 x(t + h)=x(t)+x (t) h + x (t) h2 + x (t) h3 + ... t 2 tt 6 ttt for convergence analysis and approximations Define local frame ( t , n , b ) (Frenet frame) xt xt xtt t = n = b t b = × xt × xt xtt ⇥ × ⇥ tangent main normal binormal !27 The Frenet Frame Orthonormalization of local frame x ttt b xtt n xt t local affine frame Frenet frame !28 The Frenet Frame Frenet-Serret: Derivatives w.r.t. arc length s ts =+κn ns = κt +⇥b b = − ⇥n s − Curvature (deviation from straight line) b κ = x n ss t Torsion (deviation from planarity) 1 ⇥ = det([x , x , x ]) κ2 s ss sss !29 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 b n Osculating circle! • second order contact with curve t • center c = x +(1/)n • radius 1/ !30 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 Curvature: Some Intuition A line through two points on the curve (Secant) !32 Curvature: Some Intuition A line through two points on the curve (Secant) !33 Curvature: Some Intuition Tangent, the first approximation limiting secant as the two points come together !34 Curvature: Some Intuition Circle of curvature Consider the circle passing through 3 pints of the curve !35 Curvature: Some Intuition Circle of curvature The limiting circle as three points come together !36 Curvature: Some Intuition Radius of curvature r !37 Curvature: Some Intuition Radius of curvature r !38 Curvature: Some Intuition Signed curvature Sense of traversal along curve !39 Curvature: Some Intuition Gauß map Point on curve maps to point on unit circle !40 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 Curvature: Some Intuition Turning number, k Number of orbits in Gaussian image !42 Curvature: Some Intuition Turning number theorem For a closed curve, the integral of curvature is an integer multiple of 2" !43 Take Home Message In the limit of a refinement sequence, discrete measure of length and curvature agree with continuous measures !44 Outline • Parametric Curves • Parametric Surfaces !45 Surfaces What characterizes shape?! • shape does not depend on Euclidean motions • metric and curvatures • smooth continuous notions to discrete notions !46 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 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 ⇥ !48 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 !49 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 ⇥ ⇥ !50 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⇥ !51 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 !52 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 ⇥ !53 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 !54 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 !55 Normal Curvature Tangent vector t … n xu xv p t xu xv t = cos φ + sin φ ∥xu∥ ∥xv∥ unit vector !56 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∥ !57 Geometry of the Normal Gauss map! • normal at point ! ! ! • consider curve in surface again • study its curvature at p • normal “tilts” along curve !58 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) !59 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 !60 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
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