05 - Surfacespanozzo/gp18/05 - Surfaces.pdf · 2019. 1. 31. · Usage example: define fair...
Transcript of 05 - Surfacespanozzo/gp18/05 - Surfaces.pdf · 2019. 1. 31. · Usage example: define fair...
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
05 - Surfaces
Acknowledgements: Olga Sorkine-Hornung
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Reminder – Curves
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Turning Number TheoremContinuous world Discrete world
k:
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Curvature is scale dependent
is scale-independent
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Curvature – Integrated Quantity!
• Cannot view αi as pointwise curvature
• It is integrated curvature over a local area associated with vertex i
α1 α2
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Curvature – Integrated Quantity!
• Integrated over a local area associated with vertex i α1 α2
A1
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Curvature – Integrated Quantity!
• Integrated over a local area associated with vertex i α1 α2
A1 A2
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Curvature – Integrated Quantity!
• Integrated over a local area associated with vertex i α1 α2
A1 A2
The vertex areas Ai form a covering of the curve. They are pairwise disjoint (except endpoints).
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Structure Preservation
• Arbitrary discrete curve • total signed curvature obeys
discrete turning number theorem • even coarse mesh (curve) • which continuous theorems to preserve?
• that depends on the application…
discrete analogue of continuous theorem
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Recap
ConvergenceStructure- preservation
In the limit of a refinement sequence, discrete measures of length and curvature agree with continuous measures.
For an arbitrary (even coarse) discrete curve, the discrete measure of curvature obeys the discrete turning number theorem.
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Surfaces
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Surfaces, Parametric Form• Continuous surface
• Tangent plane at point p(u,v) is spanned by
n
p(u,v)pu
u
v
pv
These vectors don’t have to be orthogonal
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Isoparametric Lines• Lines on the surface when
keeping one parameter fixed
u
v
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Surface Normals
• Surface normal:
• Assuming regular parameterization, i.e.,
n
p(u,v)
pu
u
v
pv
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
n
p
pu pv
t
Normal Curvature
Unit-length direction t in the tangent plane (if pu and pv are orthogonal):
tϕ
Tangent plane
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Normal Curvature
n
p
pu pv
tγ
The curve γ is the intersection of the surface with the plane through n and t.
Normal curvature:
κn(ϕ) = κ(γ(p))
tϕ
Tangent plane
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Surface Curvatures
• Principal curvatures • Minimal curvature
• Maximal curvature
• Mean curvature
• Gaussian curvature
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Principal Directions
• Principal directions: tangent vectors corresponding to ϕmax and ϕmin
min curvature max curvaturetangent plane
ϕ min
t1t2
Pierre Alliez, David Cohen-Steiner, Olivier Devillers, Bruno Lévy, and Mathieu Desbrun. 2003. Anisotropic polygonal remeshing. ACM Trans. Graph. 22, 3 (July 2003), 485-493. DOI: http://dx.doi.org/10.1145/882262.882296
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Euler’s Theorem: Planes of principal curvature are orthogonal and independent of parameterization.
Principal Directions
By Eric Gaba (Sting) - Based upon a drawing in a book, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1325452
https://commons.wikimedia.org/w/index.php?curid=1325452
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Principal Directions
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Mean Curvature
• Intuition for mean curvature
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Classification
• A point p on the surface is called • Elliptic, if K > 0 • Parabolic, if K = 0 • Hyperbolic, if K < 0
• Developable surface iff K = 0
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Local Surface Shape By Curvatures
Isotropic: all directions are principal directions
spherical (umbilical) planar
K > 0, κ1= κ2
Anisotropic: 2 distinct principal directions
elliptic parabolic hyperbolic
κ1 > 0, κ2 > 0
κ1= 0
κ2 > 0
K > 0 K = 0 K < 0
K = 0
κ1 < 0
κ2 > 0
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Usage example: define fair surfaces
• FiberMesh
s.t. M interpolates the curves
Andrew Nealen, Takeo Igarashi, Olga Sorkine and Marc Alexa, "FiberMesh: Designing Freeform Surfaces with 3D Curves", ACM Transactions on Computer Graphics, ACM SIGGRAPH 2007, San Diego, USA, 2007
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Gauss-Bonnet Theorem
• For a closed surface M:
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Gauss-Bonnet Theorem
• For a closed surface M:
• Compare with planar curves:
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Fundamental Forms
• First fundamental form
• Second fundamental form
• Together, they define a surface (if some compatibility conditions hold)
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Fundamental Forms
• I allows to measure • length, angles, area, curvature • arc element
• area element
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Intrinsic Geometry
• Properties of the surface that only depend on the first fundamental form • length • angles • Gaussian curvature (Theorema Egregium)
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Interesting examples
• Hyperbolic geometry – spaces with constant negative Gauss curvature
• Constant mean curvature surfaces • Minimal surfaces (H = 0)
Lobachevsky plane Soap surfaces (minimal surfaces)
By Anders Sandberg - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=21537940
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Laplace Operator
Laplace operator
gradient operator
2nd partial derivatives
Cartesian coordinatesdivergence
operator
function in Euclidean space
Intuitive Explanation
The Laplacian Δf(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p deviates from f(p) as the radius of the sphere grows.
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Laplace-Beltrami Operator• Extension of Laplace to functions on manifolds
Laplace- Beltrami
gradient operator
divergence operator
function on surface M
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Laplace-Beltrami Operator
• For coordinate functions:
mean curvature
unit surface normal
Laplace- Beltrami
gradient operator
divergence operator
function on surface M
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Differential Geometry on Meshes
• Assumption: meshes are piecewise linear approximations of smooth surfaces
• Can 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 prone
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
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
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Uniform discretization:
• Depends only on connectivity = simple and efficient
• Bad approximation for irregular triangulations
vi
vj
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Intuition for uniform discretization
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Intuition for uniform discretization
vi vi+1vi-1
γ
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
vi
vj1 vj2
vj3
vj4vj5
vj6
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Cotangent formula
Aivi
vi
vj vj
αij
βij
vi
vj
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Voronoi Vertex Area
• Unfold the triangle flap onto the plane (without distortion)
θ
vi
vj
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Voronoi Vertex Area
θ
vi
cjvj
cj+1
Flattened flapvi
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Cotangent formula
• Accounts for mesh geometry
• Potentially negative/ infinite weights
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Cotangent formula
• Can be derived using linear Finite Elements • Nice property: gives zero for planar 1-rings!
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Uniform Laplacian Lu(vi) • Cotangent Laplacian Lc(vi) • Normal
vi
vjα
β
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Uniform Laplacian Lu(vi) • Cotangent Laplacian Lc(vi) • Normal
• For nearly equal edge lengths Uniform ≈ Cotangent
vi
vjα
β
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Uniform Laplacian Lu(vi) • Cotangent Laplacian Lc(vi) • Normal
• For nearly equal edge lengths Uniform ≈ Cotangent
vi
vjα
β
Cotan Laplacian allows computing discrete normal
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Curvatures
• Mean curvature (sign defined according to normal)
• Gaussian curvature
• Principal curvatures
Ai
θj
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Gauss-Bonnet Theorem
• Total Gaussian curvature is fixed for a given topology
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Example: Discrete Mean Curvature
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Links and Literature
• M. Meyer, M. Desbrun, P. Schroeder, A. Barr Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, VisMath, 2002
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Links and Literature
https://www.cs.cmu.edu/~kmcrane/Projects/DGPDEC/
Discrete Differential Geometry: An Applied Introduction Keenan Crane
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Links and Literature
• libigl implements many discrete differential operators
• See the tutorial! • http://libigl.github.io/libigl/tutorial/tutorial.html
Principal Directions
http://libigl.github.io/libigl/tutorial/tutorial.htmlhttp://libigl.github.io/libigl/tutorial/tutorial.htmlhttp://libigl.github.io/libigl/tutorial/tutorial.htmlhttp://libigl.github.io/libigl/tutorial/tutorial.htmlhttp://libigl.github.io/libigl/tutorial/tutorial.htmlhttp://libigl.github.io/libigl/tutorial/tutorial.htmlhttp://libigl.github.io/libigl/tutorial/tutorial.html
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CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Thank you