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 x t ( t ) x ( a ) x ( t ) d x ( t ) / d t x t ( t ) := d x ( t ) x ( t ) = y ( t ) = d y ( t ) / d t d t z ( t ) d z ( t ) / d t � 19
Recall: Mappings Bijective Injective Surjective NO SELF-INTERSECTIONS � SELF-INTERSECTIONS � AMBIGUOUS PARAMETERIZATION � 20
Parametric Curves x ( t ) A parametric curve is � • simple: is injective (no self-intersections) x ( t ) t ∈ [ a, b ] x t ( t ) • differentiable: is defined for all x t ( t ) 6 = 0 t ∈ [ a, b ] • regular: for all x ( b ) x ( t ) x t ( t ) x ( a ) � 21
Length of a Curve x i = x ( t i ) Let and t i = a + i ∆ t x i x ( a ) x ( b ) t i ∆ t b a � 22
Length of a Curve Polyline chord length � � ∆ x i ⇥ ⇥ � � ⇥ ∆ x i ⇥ = ∆ x i := ⇥ x i +1 � x i ⇥ S = � ∆ t , � � ∆ t � i i norm change Curve arc length ( ) ∆ t → 0 x i � t s = s ( t ) = � x t � d t x ( a ) a x ( b ) length = � integration of infinitesimal change � t i ∆ t b a × 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) d x ( u ( t )) = d x d u d t = x u ( u ( t )) u t ( t ) d t d u � 24
Re-Parameterization Example � ⇥ 0 , 1 � IR 2 f : t ⇥� (4 t, 2 t ) , 2 � ⇥ 0 , 1 φ : � [0 , 1] t ⇥� 2 t , 2 g : [0 , 1] � IR 2 t ⇥� (2 t, t ) , g ( φ ( t )) = f ( t ) ⇒ � 25
Arc Length Parameterization Mapping of parameter domain: � t t ⇥� s ( t ) = ⇤ x t ⇤ d t a Parameter for equals length from to x ( s ) x ( s ) x ( a ) s x ( s ) = x ( s ( t )) d s = � x t � d t same infinitesimal change Special properties of resulting curve ⇥ x s ( s ) ⇥ = 1 , x s ( s ) · x ss ( s ) = 0 defines orthonormal frame � 26
The Frenet Frame Taylor expansion x ( t + h ) = x ( t ) + x t ( t ) h + 1 2 x tt ( t ) h 2 + 1 6 x ttt ( t ) h 3 + . . . for convergence analysis and approximations ( t , n , b ) Define local frame (Frenet frame) x t � x tt x t b = t = n = b × t � x t � ⇥ x t � x tt ⇥ tangent main normal binormal � 27
The Frenet Frame Orthonormalization of local frame x ttt b x tt n t x t local affine frame Frenet frame � 28
The Frenet Frame Frenet-Serret: Derivatives w.r.t. arc length s t s = + κ n n s = − κ t + τ b b s = − τ n Curvature (deviation from straight line) b n κ = � x ss � t Torsion (deviation from planarity) 1 τ = κ 2 det([ x s , x ss , x sss ]) � 29
Curvature and Torsion Planes defined by and two vectors: � x • osculating plane: vectors and t n • normal plane: vectors and b n • rectifying plane: vectors and b t b n Osculating circle � t • second order contact with curve c = x + (1 / κ ) n • center 1 / κ • radius � 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 π � 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 ) y ( u, v ) x ( 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 � 48
Angles on Surface [ u ( t ) , v ( t )] Curve in uv-plane defines curve on the surface x ( u, v ) 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 � � 49
Angles on Surface [ u ( t ) , v ( t )] Curve in uv-plane defines curve on the surface x ( u, v ) 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 � 50
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 , � 51
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