Differential GeometryMark Pauly

Mark Pauly

Outline

• Differential Geometry– curvature– fundamental forms– Laplace-Beltrami operator

• Discretization

• Visual Inspection of Mesh Quality

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Mark Pauly

Differential Geometry

• Continuous surface

• Normal vector

– assuming regular parameterization, i.e.

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x(u, v) =

!

"

x(u, v)y(u, v)z(u, v)

#

$ , (u, v) ! IR2

n = (xu ! xv)/"xu ! xv"

xu ! xv "= 0

Mark Pauly

• Normal Curvature

Differential Geometry

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n

p

n =

xu ! xv

"xu ! xv"

t

xu xv

t = cos !xu

!xu!+ sin!

xv

!xv!

Mark Pauly

• Normal Curvature

Differential Geometry

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t

n

pc

n =

xu ! xv

"xu ! xv"

t = cos !xu

!xu!+ sin!

xv

!xv!

Mark Pauly

Differential Geometry

• Principal Curvatures– maximum curvature

– minimum curvature

• Euler Theorem:

• Mean Curvature

• Gaussian Curvature

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K = !1 · !2

!n(t̄) = !n(") = !1 cos2 " + !2 sin2 "

H =!1 + !2

2=

1

2"

! 2!

0

!n(#)d#

!1 = max!

!n(")

!2 = min!

!n(")

Mark Pauly

Differential Geometry

• Normal curvature is defined as curvature of the normal curve at a point

• Can be expressed in terms of fundamental forms as

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t

n

pc

p ! cc ! x(u, v)

!n(t̄) =t̄T II t̄

t̄T I t̄=

ea2 + 2fab + gb2

Ea2 + 2Fab + Gb2

t = axu + bxv

Mark Pauly

Differential Geometry

• First fundamental form

• Second fundamental form

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I =

!

E F

F G

"

:=

!

xTu xu xT

u xv

xTu xv xT

v xv

"

II =

!

e ff g

"

:=

!

xTuun xT

uvn

xTuvn xT

vvn

"

Mark Pauly

Differential Geometry

• I and II allow to measure– length, angles, area, curvature– arc element

– area element

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ds2

= Edu2

+ 2Fdudv + Gdv2

dA =

!

EG ! F 2dudv

Mark Pauly

Differential Geometry

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

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K = limr!0

6!r ! 3C(r)

!r3

Mark Pauly

• A point x on the surface is called– elliptic, if K > 0– parabolic, if K = 0– hyperbolic, if K < 0– umbilical, if

• Developable surface ⇔ K = 0

Differential Geometry

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!1 = !2

Mark Pauly

Laplace Operator

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!f = div!f =!

i

!2f

!x2i

Cartesiancoordinates

divergenceoperator

gradientoperator

Laplaceoperator

function in Euclidean space

2nd partialderivatives

Mark Pauly

Laplace-Beltrami Operator

• Extension of Laplace to functions on manifolds

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divergenceoperator

gradientoperator

Laplace-Beltrami

function onmanifold

!Sf = divS !Sf

S

Mark Pauly

Laplace-Beltrami Operator

• Extension of Laplace to functions on manifolds

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surfacenormal

meancurvature

divergenceoperator

gradientoperator

Laplace-Beltrami

coordinatefunction

!Sx = divS !Sx = "2Hn

Mark Pauly

Outline

• Differential Geometry– curvature– fundamental forms– Laplace-Beltrami operator

• Discretization

• Visual Inspection of Mesh Quality

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Mark Pauly

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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Mark Pauly

Discrete Laplace-Beltrami

• Uniform discretization

– depends only on connectivity → simple and efficient

– bad approximation for irregular triangulations

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!unif (v) :=1

|N1 (v)|

!

vi!N1(v)

(f (vi) ! f (v))

Mark Pauly

Discrete Laplace-Beltrami

• Cotangent formula

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!Sf(v) :=2

A(v)

!

vi!N1(v)

(cot !i + cot "i) (f(vi) ! f(v))

v

vi vi

vA(v) v

vi

!i

!i

Mark Pauly

Discrete Laplace-Beltrami

• Cotangent formula

• Problems– negative weights– depends on triangulation

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!Sf(v) :=2

A(v)

!

vi!N1(v)

(cot !i + cot "i) (f(vi) ! f(v))

Mark Pauly

Discrete Curvatures

• Mean curvature

• Gaussian curvature

• Principal curvatures

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G = (2! !

!

j

"j)/A

A

!j

!1 = H +

!

H2! G !2 = H !

!

H2! G

H = !!Sx!

Mark Pauly

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

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Mark Pauly

Outline

• Differential Geometry– curvature– fundamental forms– Laplace-Beltrami operator

• Discretization

• Visual Inspection of Mesh Quality

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Mark Pauly

Mesh Quality

• Smoothness– continuous differentiability of a surface (Ck)

• Fairness– aesthetic measure of “well-shapedness”– principle of simplest shape– fairness measures from physical models

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!S

!2

1 + !2

2 dA

!

S

"

!"1

!t1

#2

+

"

!"2

!t2

#2

dA

strain energy variation of curvature

Mark Pauly

Mesh Quality

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!S

!2

1 + !2

2 dA

!

S

"

!"1

!t1

#2

+

"

!"2

!t2

#2

dA

strain energy variation of curvature

Mark Pauly

Mesh Quality

• Visual inspection of “sensitive” attributes– Specular shading

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Mark Pauly

Mesh Quality

• Visual inspection of “sensitive” attributes– Specular shading– Reflection lines

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Mark Pauly

Mesh Quality

• Visual inspection of “sensitive” attributes– Specular shading– Reflection lines

• differentiability one order lower than surface• can be efficiently computed using graphics hardware

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C0

C1

C2

Mark Pauly

Mesh Quality

• Visual inspection of “sensitive” attributes– Specular shading– Reflection lines– Curvature

• Mean curvature

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Mark Pauly

Mesh Quality

• Visual inspection of “sensitive” attributes– Specular shading– Reflection lines– Curvature

• Mean curvature• Gauss curvature

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Mark Pauly

• Smoothness– Low geometric noise

Mesh Quality Criteria

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Mark Pauly

Mesh Quality Criteria

• Smoothness– Low geometric noise

• Adaptive tessellation– Low complexity

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Mark Pauly

Mesh Quality Criteria

• Smoothness– Low geometric noise

• Adaptive tessellation– Low complexity

• Triangle shape– Numerical robustness

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Mark Pauly

• Circum radius / shortest edge

• Needles and caps

Triangle Shape Analysis

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Needle Cap

r1

e1r2

e2r1

e1

<r2

e2

Mark Pauly

Mesh Quality Criteria

• Smoothness– Low geometric noise

• Adaptive tessellation– Low complexity

• Triangle shape– Numerical robustness

• Feature preservation– Low normal noise

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Mark Pauly

Normal Noise Analysis

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Mark Pauly

Mesh Optimization

• Smoothness➡ Mesh smoothing

• Adaptive tessellation➡ Mesh decimation

• Triangle shape➡ Repair, remeshing

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