(Discrete) Differential Geometry Motivation • Understand the structure of the surface – Properties: smoothness, “curviness”, important directions • How to modify the surface to change these properties • What properties are preserved for different modifications • The math behind the scenes for many geometry processing applications 2 Motivation • Smoothness ➡ Mesh smoothing • Curvature ➡ Adaptive simplification ➡ Parameterization 3 Motivation • Triangle shape ➡ Remeshing • Principal directions ➡ Quad remeshing 4 Differential Geometry • M.P. do Carmo: Differential Geometry of Curves and Surfaces, Prentice Hall, 1976 Leonard Euler (1707 - 1783) Carl Friedrich Gauss (1777 - 1855) 5 Parametric Curves “velocity” of particle on trajectory 6 Parametric Curves A Simple Example α1(t)() = (a cos(t), a s(sin(t)) t t ∈ [0,2π] a α2(t) = (a cos(2t), a sin(2t)) t ∈ [0,π] 7 Parametric Curves A Simple Example α'(t) α(t))( = (cos(t), s in (t)) t α'(t) = (-sin(t), cos(t)) α'(t) - direction of movement ⏐α'(t)⏐ - speed of movement 8 Parametric Curves A Simple Example α2'(t) α1'(t) α (t)() = (cos(t), sin (t)) t 1 α2(t) = (cos(2t), sin(2t)) Same direction, different speed 9 Length of a Curve • Let and 10 Length of a Curve • Chord length Euclidean norm 11 Length of a Curve • Chord length • Arc length 12 Examples y(t) x(t) • Straight line – x(t)=() = (t,t), t∈[0,∞) – x(t) = (2t,2t), t∈[0,∞) – x(t) = (t2,t2), t∈[0,∞) x(t) • Circle y(t) – x(t) = (cos(t),sin(t)), t∈[0,2π) x(t) x(t) 13 Examples α(t) = (a cos(t), a sin(t)), t ∈ [0,2π] α'(t) = (-a sin(t), a cos(t)) Many possible parameterizations Length of the curve does not depend on parameterization! 14 Arc Length Parameterization • Re‐parameterization • Arc length parameterization – parameter value s for x(s) equals length of curve from x(a) to x(s) 15 Curvature x(t) a curve parameterized by arc length The curvature of x at t: – the tangent vector at t – the change in the tangent vector at t R(t) = 1/κ (t) is the radius of curvature at t 16 Examples Straight line α(s)=) = us + v, uvu,v∈ R2 α'(s) = u α''(s) = 0 → ⏐α''(s)⏐ = 0 Circle α(s) = (a cos(s/a), a sin(s/a)), s ∈ [0,2πa] α'(s) = (-sin(s/a), cos(s/a)) α''(s)() = (-cos(s/a)/a, -si(in(s/a)/a) → ⏐α''(s)⏐ = 1/a 17 The Normal Vector x'(t) x'(t) = T(t) ‐ tangent vector |x'(t)| ‐ arc length x''(t) x''(t)=) = T'(t) ‐ normal direction |x''(t)| ‐ curvature x'(t) If |x''(t)| ≠ 0, define N(t) = T'(t)/|T'(t)| x''(t) Then x''(t) = T'(t) = κ(t)N(t)18 The Osculating Plane The plane determined by the unit tangent and normal vectors T T(s) and N(s) is called the N osculating plane at s N T 19 The Binormal Vector For points s, s.t. κ(s) ≠ 0, the binormal vector B(s) is defined as: B(s) = T(s) × N(s) T B B N N The binormal vector defines the osculating plane T 20 The Frenet Frame tangent normal binormal 21 The Frenet Frame {T(s), N(s), B(s)} form an orthonormal basis for R3 calle d the Frenet frame How does the frame change when the particle moves? Wha t are T', N', B' in terms of T, N, B ? 22 The Frenet Frame • Frenet‐Serret formulas • curvature • torsion (arc-lenggpth parameterization ) 23 Curvature and Torsion • Curvature: Deviation from straight line • Torsion: Deviation from planarity • Independent of parameterization – intrinsic properties of the curve • Invariant under rigid (li(translation+rotat i)ion) motion • Define curve uniquely up to rigid motion 24 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 x • Osculilating cilircle – second order contact with curve – center – radius C R 25 Differential Geometry: Surfaces 26 Differential Geometry: Surfaces • Continuous surface • NlNormal vector – assuming regular pp,arameterization, i.e. 27 Normal Curvature If xxuv and are orthogonal: 28 Normal Curvature 29 Surface Curvature • Principal Curvatures – maximum curvature – minimum curvature • Mean Curvature • Gaussian Curvature 30 Principal Curvature Euler’s Theorem: Planes of principal curvature are orthogonal and independent of parameterization. 31 Curvature 32 Surface Classification k1= k2 > 0 k1 = k2= 0 Isotropic Equal in all directions spherical planar k2 = 0 k > 0 k2 < 0 Anisotropic 2 k1 > 0 k1> 0 k > 0 Distinct principal 1 directions elliptic parabolic hyperbolic K > 0 K = 0 K < 0 dlbldevelopable 33 Principal Directions k 2 k1 34 Gauss‐Bonnet Theorem For ANY closed manifold surface with Euler number χ=2-2g: ∫K()( ) =∫K()( ) =∫K()( ) =4π 35 Gauss‐Bonnet Theorem Example Sphere k1 = k2= 1/r 2 K = k1k2 = 1/r Manipulate sphere New positive + negative curvature ClCancel out! 36 Fundamental Forms • First fundamental form • Second fundamental form 37 Fundamental Forms • I and II allow to measure – length, angles, area, curvature – arc element – area element 38 Fundamental Forms • Normal curvature = curvature of the normal curve at point • Can be expressed in terms of fundamental forms as 39 Intrinsic Geometry • Properties of the surface that only depend on the first fundamental form – length – angles – Gaussian curvature (Theorema Egregium) 40 Laplace Operator gradient Laplace operator 2nd partial operator derivatives ∈ R Cartesian scalar function in divergence coordinates Euclidean space operator 41 Laplace‐Beltrami Operator • Extension to functions on manifolds Laplace- gradient Beltrami operator scalar function divergence on manifold operator 42 Laplace‐Beltrami Operator • For coordinate function(s) Laplace- gradient operator mean BltBeltram i curvature surface coordinate divergence normal function operator 43 Discrete Differential Operators • Assumption: Meshes are piecewise linear approximations of smooth surfaces • Approach: Approximate differential ppproperties at point v as finite differences over local mesh neighborhood N(v) v – v = mesh vertex – Nd(v) = d‐ring neighborhood N1(v) • Disclaimer: many possible discretizations, none is “perfect” 44 Functions on Meshes • Function f given at mesh vertices f(vi) = f(xi) = fi • Linear interpolation to triangle 1 Bi (x) xk xi xj 45 Gradient of a Function 1 Steepest ascent direction Bi (x) perpendicu lar to opposite edge xk xi xj Constant in the triangle 46 Gradient of a Function 47 Discrete Laplace‐Beltrami First Approach • Laplace operator: • In 2D: • On a grid – finite differences discretization: 48 Discrete Laplace‐Beltrami Uniform Discretization v N1(v) Normalized: 49 Discrete Laplace‐Beltrami Second Approach • Laplace‐Beltrami operator: • Compute integral around vertex Divergence theorem 50 Discrete Laplace‐Beltrami Cotangent Formula Plugging in expression for gradients gives: 51 The Averaging Region Edge midpoint θ Barycentric cell Voronoi cell Mixed cell ci = btbarycenter ci = citircumcenter of triangle of triangle 52 The Averaging Region Mixed Cell If θ < π/2, ci is the circumcenter of the triangle (vi , v, vi+1 ) θ If θ ≥ π/2, ci is the midpo int of the edge (vi , vi+1 ) 53 Discrete Normal n()x 54 Discrete Curvatures • Mean curvature • Gaussian curvature • Principal curvatures 55 Example: Mean Curvature 56 Example: Gaussian Curvature original smoothed Discrete Gauss-Bonnet (Descartes) theorem: 57 References • “Discrete Differential‐Geometry Operators for Triangulated 2‐ Manifolds”, Meyer et al., ’02 • “Restricted Delaunay triangulations and normal cycle”, Cohen‐Steiner et al., SoCG ‘03 • “On the convergence of metric and geometric properties of polyhedral surfaces”, Hildebrandt et al., ’06 • “Discrete Laplace operators: No free lunch”, Wardetzky et al., SGP ‘07 58.