# 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

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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

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Differential Geometry
• Normal Curvature

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

xu         xv
t = cos φ    + sin φ
xu         xv

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Differential Geometry
• Normal Curvature

xu × xv
n                        n=
xu × x v

t                       c
p

xu         xv
t = cos φ    + sin φ
xu         xv

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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

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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

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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

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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

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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

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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

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Laplace Operator

Laplace                                 2nd partial
operator
operator                                derivatives

∂2f
∆f = div f =
i
∂x2
i

function in                               Cartesian
divergence
Euclidean space                            coordinates
operator

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Laplace-Beltrami Operator
• Extension of Laplace to functions on manifolds

Laplace-
operator
Beltrami

∆S f = divS      Sf

function on
divergence
manifold S
operator

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Laplace-Beltrami Operator
• Extension of Laplace to functions on manifolds

Laplace-                                    mean
operator
Beltrami                                  curvature

∆S x = divS      Sx   = −2Hn

coordinate                                   surface
divergence             normal
function
operator

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

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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

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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

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• 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

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

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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
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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

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Mesh Quality
• Visual inspection of “sensitive” attributes
– Reﬂection lines

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

C0                C1             C2
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Mesh Quality
• Visual inspection of “sensitive” attributes
– Reﬂection lines
– Curvature
• Mean curvature

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Mesh Quality
• Visual inspection of “sensitive” attributes
– Reﬂection lines
– Curvature
• Mean curvature
• Gauss curvature

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Mesh Quality Criteria
• Smoothness
– Low geometric noise

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Mesh Quality Criteria
• Smoothness
– Low geometric noise
– Low complexity

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Mesh Quality Criteria
• Smoothness
– Low geometric noise
– Low complexity
• Triangle shape
– Numerical robustness

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Triangle Shape Analysis
• Circum radius / shortest edge

r1            r1   r2
<                          e2
e1   e2
e1                               r2

• Needles and caps

Needle                 Cap

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Mesh Quality Criteria
• Smoothness
– Low geometric noise
– Low complexity
• Triangle shape
– Numerical robustness
• Feature preservation
– Low normal noise

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Normal Noise Analysis

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Mesh Optimization
• Smoothness
➡ Mesh smoothing

➡ Mesh decimation

• Triangle shape
➡ Repair, remeshing

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 views: 11 posted: 11/27/2011 language: English pages: 36