Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
05 - Surfaces
Acknowledgements: Olga Sorkine-Hornung
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Reminder – Curves
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Turning Number TheoremContinuous world Discrete world
k:
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Curvature is scale dependent
is scale-independent
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
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Curvature – Integrated Quantity!
• Integrated over a local area associated with vertex i α1 α2
A1
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
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).
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
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.
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Surfaces
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
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Isoparametric Lines• Lines on the surface when
keeping one parameter fixed
u
v
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
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
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
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Surface Curvatures
• Principal curvatures • Minimal curvature
• Maximal curvature
• Mean curvature
• Gaussian curvature
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
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
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Principal Directions
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Mean Curvature
• Intuition for mean curvature
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
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
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
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Gauss-Bonnet Theorem
• For a closed surface M:
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Gauss-Bonnet Theorem
• For a closed surface M:
• Compare with planar curves:
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)
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Fundamental Forms
• I allows to measure • length, angles, area, curvature • arc element
• area element
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)
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
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.
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
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
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
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
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
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Intuition for uniform discretization
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Intuition for uniform discretization
vi vi+1vi-1
γ
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
vi
vj1 vj2
vj3
vj4vj5
vj6
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Cotangent formula
Aivi
vi
vj vj
αij
βij
vi
vj
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Voronoi Vertex Area
• Unfold the triangle flap onto the plane (without distortion)
θ
vi
vj
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Voronoi Vertex Area
θ
vi
cjvj
cj+1
Flattened flapvi
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Cotangent formula
• Accounts for mesh geometry
• Potentially negative/ infinite weights
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!
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Laplace-Beltrami
• Uniform Laplacian Lu(vi) • Cotangent Laplacian Lc(vi) • Normal
vi
vjα
β
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
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
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Curvatures
• Mean curvature (sign defined according to normal)
• Gaussian curvature
• Principal curvatures
Ai
θj
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Discrete Gauss-Bonnet Theorem
• Total Gaussian curvature is fixed for a given topology
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Example: Discrete Mean Curvature
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
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
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
CSCI-GA.3033-018 - Geometric Modeling - Daniele Panozzo
Thank you