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Differential Geometry - PDF

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					Differential Geometry
       Mark Pauly
Outline
• Differential Geometry
  – curvature
  – fundamental forms
  – Laplace-Beltrami operator
• Discretization
• Visual Inspection of Mesh Quality




                        Mark Pauly    2
Differential Geometry
• Continuous surface
                                     
                     x(u, v)
         x(u, v) =  y(u, v)  , (u, v) ∈ I 2
                                          R
                     z(u, v)

• Normal vector
              n = (xu × xv )/ xu × xv

  – assuming regular parameterization, i.e.
                       xu × xv = 0

                         Mark Pauly             3
Differential Geometry
• Normal Curvature

                                                 xu × xv
                     n                        n=
                                                 xu × x v
              xu               xv
                     p
             t

                                                xu         xv
                                      t = cos φ    + sin φ
                                                xu         xv


                         Mark Pauly                         4
Differential Geometry
• Normal Curvature

                                                 xu × xv
                     n                        n=
                                                 xu × x v

               t                       c
                     p


                                                xu         xv
                                      t = cos φ    + sin φ
                                                xu         xv


                         Mark Pauly                         5
Differential Geometry
• Principal Curvatures
  – maximum curvature κ1 = max κn (φ)
                                      φ

  – minimum curvature κ2 = min κn (φ)
                                      φ


• Euler Theorem: κn (¯ = κn (φ) = κ1 cos2 φ + κ2 sin2 φ
                     t)

                                              2π
                        κ1 + κ2    1
• Mean Curvature     H=         =                  κn (φ)dφ
                           2      2π      0

• Gaussian Curvature K = κ1 · κ2

                         Mark Pauly                           6
Differential Geometry
• Normal curvature is defined as curvature of the
  normal curve c ∈ x(u, v) at a point p ∈ c
• Can be expressed in terms of fundamental forms
  as
               ¯ II ¯
                T           2
                        ea + 2f ab + gb2
           ¯ = t
       κn (t)
                     t
               ¯T I ¯ = Ea2 + 2F ab + Gb2
                t t

                            n

                        t   p
                                   c   t = axu + bxv



                      Mark Pauly                       7
Differential Geometry
• First fundamental form

                E   F            xT xu
                                  u       x T xv
                                            u
         I=             :=
                F   G            xT xv
                                  u       x T xv
                                            v


• Second fundamental form

                e   f                xT n xT n
                                      uu   uv
         II =           :=
                f   g                xT n xT n
                                      uv   vv




                        Mark Pauly                 8
Differential Geometry
• I and II allow to measure
  – length, angles, area, curvature
  – arc element
               ds2 = Edu2 + 2F dudv + Gdv 2
  – area element
                  dA =   EG − F 2 dudv




                         Mark Pauly           9
Differential Geometry
• Intrinsic geometry: Properties of the surface that
  only depend on the first fundamental form
  – length
  – angles
  – Gaussian curvature (Theorema Egregium)

                        6πr − 3C(r)
                K = lim
                    r→0     πr3



                       Mark Pauly                  10
Differential Geometry
• A point x on the surface is called
  – elliptic, if K > 0
  – parabolic, if K = 0
  – hyperbolic, if K < 0
  – umbilical, if κ1 = κ2


• Developable surface            K=0



                            Mark Pauly   11
Laplace Operator

                       gradient
     Laplace                                 2nd partial
                       operator
     operator                                derivatives

                                       ∂2f
                ∆f = div f =
                                   i
                                       ∂x2
                                         i


    function in                               Cartesian
                       divergence
  Euclidean space                            coordinates
                        operator



                      Mark Pauly                           12
Laplace-Beltrami Operator
• Extension of Laplace to functions on manifolds

                            gradient
      Laplace-
                            operator
      Beltrami


                 ∆S f = divS      Sf



      function on
                            divergence
      manifold S
                             operator


                          Mark Pauly               13
Laplace-Beltrami Operator
• Extension of Laplace to functions on manifolds

                            gradient
      Laplace-                                    mean
                            operator
      Beltrami                                  curvature


                 ∆S x = divS      Sx   = −2Hn


      coordinate                                   surface
                            divergence             normal
       function
                             operator


                          Mark Pauly                         14
Outline
• Differential Geometry
  – curvature
  – fundamental forms
  – Laplace-Beltrami operator
• Discretization
• Visual Inspection of Mesh Quality




                        Mark Pauly    15
Discrete Differential Operators
• Assumption: Meshes are piecewise linear
  approximations of smooth surfaces
• Approach: Approximate differential properties at
  point x as spatial average over local mesh
  neighborhood N(x), where typically
  – x = mesh vertex
  – N(x) = n-ring neighborhood or local geodesic ball




                        Mark Pauly                      16
Discrete Laplace-Beltrami
• Uniform discretization

                      1
     ∆uni f (v) :=                         (f (vi ) − f (v))
                   |N1 (v)|
                              vi ∈N1 (v)

  – depends only on connectivity → simple and
    efficient
  – bad approximation for irregular triangulations




                        Mark Pauly                             17
Discrete Laplace-Beltrami
• Cotangent formula

              2
 ∆S f (v) :=                     (cot αi + cot βi ) (f (vi ) − f (v))
             A(v)
                    vi ∈N1 (v)



              v
                                  v       A(v)    αi         v


             vi                                         βi
                                   vi            vi


                             Mark Pauly                             18
Discrete Laplace-Beltrami
• Cotangent formula

              2
 ∆S f (v) :=                     (cot αi + cot βi ) (f (vi ) − f (v))
             A(v)
                    vi ∈N1 (v)




• Problems
  – negative weights
  – depends on triangulation


                             Mark Pauly                             19
Discrete Curvatures
• Mean curvature
    H = ∆S x

• Gaussian curvature
                                                          A
    G = (2π −       θj )/A
                j
                                                     θj
• Principal curvatures
    κ1 = H +    H2 − G                    κ2 = H −   H2 − G


                             Mark Pauly                       20
Links & Literature
• P. Alliez: Estimating Curvature Tensors
  on Triangle Meshes (source code)
   – http://www-sop.inria.fr/geometrica/team/
     Pierre.Alliez/demos/curvature/



• Wardetzky, Mathur, Kaelberer,
  Grinspun: Discrete Laplace Operators:
  No free lunch, SGP 2007


                                                principal directions




                                   Mark Pauly                    21
Outline
• Differential Geometry
  – curvature
  – fundamental forms
  – Laplace-Beltrami operator
• Discretization
• Visual Inspection of Mesh Quality




                        Mark Pauly    22
Mesh Quality
• Smoothness
  – continuous differentiability of a surface (Ck)

• Fairness
  – aesthetic measure of “well-shapedness”
  – principle of simplest shape
  – fairness measures from physical models
                                             2             2
                                      ∂κ1            ∂κ2
          κ2 + κ2 dA
           1    2                                +             dA
      S                          S    ∂t1            ∂t2
      strain energy                   variation of curvature
                         Mark Pauly                             23
Mesh Quality




                                        2             2
                                 ∂κ1            ∂κ2
       κ2 + κ2 dA
        1    2                              +             dA
   S                        S    ∂t1            ∂t2
   strain energy                 variation of curvature
                    Mark Pauly                             24
Mesh Quality
• Visual inspection of “sensitive” attributes
   – Specular shading




                        Mark Pauly              25
Mesh Quality
• Visual inspection of “sensitive” attributes
   – Specular shading
   – Reflection lines




                        Mark Pauly              26
Mesh Quality
• Visual inspection of “sensitive” attributes
   – Specular shading
   – Reflection lines
      • differentiability one order lower than surface
      • can be efficiently computed using graphics hardware




                 C0                C1             C2
                            Mark Pauly                       27
Mesh Quality
• Visual inspection of “sensitive” attributes
   – Specular shading
   – Reflection lines
   – Curvature
      • Mean curvature




                         Mark Pauly             28
Mesh Quality
• Visual inspection of “sensitive” attributes
   – Specular shading
   – Reflection lines
   – Curvature
      • Mean curvature
      • Gauss curvature




                          Mark Pauly            29
Mesh Quality Criteria
• Smoothness
  – Low geometric noise




                      Mark Pauly   30
Mesh Quality Criteria
• Smoothness
  – Low geometric noise
• Adaptive tessellation
  – Low complexity




                          Mark Pauly   31
Mesh Quality Criteria
• Smoothness
  – Low geometric noise
• Adaptive tessellation
  – Low complexity
• Triangle shape
  – Numerical robustness




                          Mark Pauly   32
Triangle Shape Analysis
• Circum radius / shortest edge

         r1            r1   r2
                          <                          e2
                       e1   e2
          e1                               r2


• Needles and caps

              Needle                 Cap


                        Mark Pauly              33
Mesh Quality Criteria
• Smoothness
  – Low geometric noise
• Adaptive tessellation
  – Low complexity
• Triangle shape
  – Numerical robustness
• Feature preservation
  – Low normal noise

                          Mark Pauly   34
Normal Noise Analysis




             Mark Pauly   35
Mesh Optimization
• Smoothness
  ➡ Mesh smoothing


• Adaptive tessellation
  ➡ Mesh decimation


• Triangle shape
  ➡ Repair, remeshing



                          Mark Pauly   36

				
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posted:11/27/2011
language:English
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