Spring 2019 CSCI 621: Digital Geometry Processing 2.2 Classic Differential Geometry 1 Hao Li http://cs621.hao-li.com 1
Spring 2018 CSCI 621: Digital Geometry Processing 2.2 Classic Differential Geometry 1 With a Twist! Hao Li http://cs621.hao-li.com 2
Some Updates: run.usc.edu/vega Another awesome free library with half-edge data-structure By Prof. Jernej Barbic 3
FYI MeshLab Popular Mesh Processing Software (meshlab.sourceforge.net) 4
FYI BeNTO3D Mesh Processing Framework for Mac (www.bento3d.com) 5
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 6
Differential Geometry Why do we care? • Geometry of surfaces • Mother tongue of physical theories • Computation: processing / simulation 7
Motivation We need differential geometry to compute • surface curvature • parameterization distortion • deformation energies 8
Applications: 3D Reconstruction 9
Applications: Head Modeling 10
Applications: Facial Animation 11
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 12
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 13
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 14
Getting Started Invariant descriptors • quantities invariant under a set of transformations Intrinsic descriptor • quantities which do not depend on a coordinate frame 15
Outline • Parametric Curves • Parametric Surfaces Formalism & Intuition 16
Differential Geometry Leonard Euler (1707-1783) Carl Friedrich Gauss (1777-1855) 17
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 18
Recall: Mappings Bijective Injective Surjective NO SELF-INTERSECTIONS SELF-INTERSECTIONS AMBIGUOUS PARAMETERIZATION 19
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 ) 20
Length of a Curve x i = x ( t i ) t i = a + i ∆ t Let and x i x ( a ) x ( b ) t i ∆ t b a 21
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 22
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 23
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 ) ⇒ 24
Arc Length Parameterization Mapping of parameter domain: � t t ⇥� s ( t ) = ⇤ x t ⇤ d t a x ( s ) Parameter for equals length from to 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 25
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 26
The Frenet Frame Orthonormalization of local frame x ttt b x tt n t x t local affine frame Frenet frame 27
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 ]) 28
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 • center c = x + (1 / κ ) n 1 / κ • radius 29
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 30
Curvature: Some Intuition A line through two points on the curve (Secant) 31
Curvature: Some Intuition A line through two points on the curve (Secant) 32
Curvature: Some Intuition Tangent, the first approximation limiting secant as the two points come together 33
Curvature: Some Intuition Circle of curvature Consider the circle passing through 3 pints of the curve 34
Curvature: Some Intuition Circle of curvature The limiting circle as three points come together 35
Curvature: Some Intuition Radius of curvature r 36
Curvature: Some Intuition Radius of curvature r 37
Curvature: Some Intuition Signed curvature Sense of traversal along curve 38
Curvature: Some Intuition Gauß map Point on curve maps to point on unit circle 39
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) 40
Curvature: Some Intuition Turning number, k Number of orbits in Gaussian image 41
Curvature: Some Intuition Turning number theorem For a closed curve, the integral of curvature is an integer multiple of 2 π 42
Take Home Message In the limit of a refinement sequence , discrete measure of length and curvature agree with continuous measures 43
http://cs621.hao-li.com Thanks! 44
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