Surfaces 2D3D Shape Manipulation 3D Printing CS 6501 Slides from Olga Sorkine Eitan Grinspun Surfaces Parametric Form Continuous surface Tangent plane at point p uv is spanned by ID: 254226
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Slide1
Discrete Differential GeometrySurfaces
2D/3D Shape Manipulation,3D Printing
CS 6501
Slides from Olga
Sorkine
,
Eitan
GrinspunSlide2
Surfaces, Parametric Form
Continuous surface
Tangent plane at point p(
u,v
)
is spanned by
n
p
(
u,v
)
p
u
u
v
p
v
2Slide3
Isoparametric Lines
Lines on the surface when keeping one parameter fixed
u
v
3Slide4
Surface Normals
Surface normal:
Assuming regular parameterization, i.e.,
n
p
(
u,v
)
p
u
u
v
p
v
4Slide5
Normal Curvature
Direction
t in the tangent plane (if pu and
p
v
are orthogonal):
t
n
p
p
u
p
v
t
Tangent plane
5Slide6
Normal Curvature
n
p
p
u
p
v
t
The curve
is the intersection
of the surface with the plane
through
n
and
t
.
Normal curvature:
n() = ((p))tTangent plane6Slide7
Surface Curvatures
Principal curvaturesMaximal curvatureMinimal curvature
Mean curvatureGaussian curvature
7Slide8
Principal Directions
Principal directions:tangent vectorscorresponding to
max and min
min curvature
max curvature
tangent plane
t
1
max
t2
8Slide9
Euler
’
s Theorem:
Planes of principal curvature are
orthogonal
and independent of parameterization.
Principal Directions
9Slide10
Principal Directions
10Slide11
Mean Curvature
Intuition for mean curvature
11Slide12
Classification
A point
p on the surface is calledElliptic, if K
> 0
Parabolic, if
K
= 0Hyperbolic, if K < 0Umbilical, ifDevelopable surface iff K = 0
12Slide13
Local Surface Shape By Curvatures
Isotropic:
all directions are
principal directions
spherical (umbilical)
planar
K
> 0,
1
=
2
Anisotropic:
2 distinct principal directions
elliptic
parabolic
hyperbolic
2 > 0, 1 > 02 = 01 > 02 < 01 > 0
K > 0
K = 0
K < 0K = 0
13Slide14
Gauss-Bonnet Theorem
For a closed surface M:
14Slide15
Gauss-Bonnet Theorem
For a closed surface M:
Compare with planar curves:
15Slide16
Fundamental Forms
First fundamental form
Second fundamental form
Together, they define a surface (given some compatibility conditions)
16Slide17
Fundamental Forms
I
and II allow to measurelength, angles, area, curvature
arc element
area element
17Slide18
Intrinsic Geometry
Properties of the surface that only depend on the first fundamental form
lengthanglesGaussian curvature (Theorema
Egregium
)
18Slide19
Laplace Operator
Laplaceoperator
gradientoperator
2nd partial
derivatives
Cartesian
coordinatesdivergenceoperatorfunction inEuclidean space
19Slide20
Laplace-Beltrami Operator
Extension of Laplace to functions on manifolds
Laplace-Beltrami
gradient
operator
divergence
operatorfunction onsurface M
20Slide21
Laplace-Beltrami Operator
mean curvature
unit
surface
normal
Laplace-
Beltramigradientoperator
divergenceoperatorfunction onsurface M
For coordinate functions:
21Slide22
Differential Geometry on Meshes
Assumption: meshes are piecewise linear approximations of smooth surfacesCan try fitting a smooth surface locally (say, a polynomial) and find differential quantities analytically
But: it is often too slow for interactive setting and error prone22Slide23
Discrete Differential Operators
Approach: approximate differential properties at point v as spatial average over local mesh neighborhood
N(v
)
where typically
v = mesh vertex
Nk(v) = k-ring neighborhood
23Slide24
Discrete Laplace-Beltrami
Uniform discretization: L
(v) or
∆
v
Depends only on connectivity
= simple and efficientBad approximation for irregular triangulations
v
i
v
j
24Slide25
Discrete Laplace-Beltrami
Intuition for uniform discretization
25Slide26
Discrete Laplace-Beltrami
Intuition for uniform discretization
v
i
v
i
+1
v
i
-1
26Slide27
Discrete Laplace-Beltrami
v
i
v
j
1
v
j
2
v
j
3
v
j
4
v
j
5
v
j6
Intuition for uniform discretization
27Slide28
Discrete Laplace-Beltrami
Cotangent formula
A
i
v
i
v
i
v
j
v
j
ij
ij
v
i
v
j28Slide29
Voronoi
Vertex Area
Unfold the triangle flap onto the plane (without distortion)
29
θ
v
i
v
jSlide30
Voronoi
Vertex Area
θ
v
i
c
j
v
j
c
j
+1
30
Flattened flap
v
iSlide31
Discrete Laplace-Beltrami
Cotangent formulaAccounts for mesh
geometryPotentially negative/infinite weights
31Slide32
Discrete Laplace-Beltrami
Cotangent formulaCan be derived using linear Finite Elements
Nice property: gives zero for planar 1-rings!
32Slide33
Discrete Laplace-Beltrami
Uniform
Laplacian
L
u
(
vi)Cotangent Laplacian Lc(vi
)Mean curvature normal
v
i
v
j
a
b
33Slide34
Discrete Laplace-Beltrami
v
i
v
j
a
b
Uniform
Laplacian
L
u
(
v
i
)
Cotangent
Laplacian
L
c
(vi)Mean curvature normalFor nearly equal edge lengthsUniform ≈ Cotangent34Slide35
Discrete Laplace-Beltrami
v
i
v
j
a
b
Uniform
Laplacian
L
u
(
v
i
)
Cotangent
Laplacian
L
c
(vi)Mean curvature normalFor nearly equal edge lengthsUniform ≈ CotangentCotan Laplacian allows computing discrete normal35Slide36
Discrete Curvatures
Mean curvature (sign defined according to normal)Gaussian curvature
Principal curvatures
A
i
j
36Slide37
Discrete Gauss-Bonnet Theorem
Total Gaussian curvature is fixed for a given topology
37Slide38
Example: Discrete Mean Curvature
38Slide39
Links and Literature
M. Meyer, M. Desbrun, P. Schroeder, A. Barr
Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, VisMath, 2002
39Slide40
P.
Alliez, Estimating Curvature Tensors on Triangle Meshes, Source Codehttp://www-sop.inria.fr/geometrica/team/Pierre.Alliez/demos/curvature/
principal directions
Links and Literature
40Slide41
Measuring Surface Smoothness
41Slide42
Links and Literature
Grinspun et al.:Computing
discrete shape operators on general meshes, Eurographics 2006
42Slide43
Reflection Lines as an Inspection Tool
Shape optimization using reflection lines
E. Tosun, Y. I. Gingold, J. Reisman, D. ZorinSymposium on Geometry Processing 2007
43Slide44
Shape optimization using reflection lines
E.
Tosun, Y. I. Gingold, J. Reisman, D. Zorin
Symposium on Geometry Processing 2007
Reflection Lines as an Inspection Tool
44Slide45
Thank You