Geometry Processing: A First Set of Applications
Peter Schröder
DDG Course SIGGRAPH 2005
1 In this Section
What characterizes shape? Q brief recall of classic notions Q mean and Gaussian curvature Q how to express them in the discrete setting of meshes? Q putting them to work Q smoothing Q parameterization DDG Course SIGGRAPH 2005
2 Surfaces
Basic setup Q parameterized surface
Q tangent vectors
Q tangent space
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3 Metric on Surface
Measure stuff Q angle, length, area Q all require an inner product Q we have: Q Euclidean inner product in domain Q want to turn this into: Q inner product on surface
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4 Lifting onto Surface
Use basis vectors Q du and dv in domain
Q S,u and S,v on surface Q record all their inner products
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5 Lifting onto Surface
Use basis vectors Q du and dv in domain
Q S,u and S,v on surface Q record all their inner products
principal “stretches”
Q two invariants: trace & determinant Q average length & area DDG Course SIGGRAPH 2005
6 First Fundamental Form
Measuring area Q areas in tangent space
Q no dependence on parameterization Q discrete setting… easy Q sum areas of triangles
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Gauss map Q normal at point
Q consider curve in surface again Q study its curvature at p Q normal “tilts” along curve
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8 Shape Operator
Derivative of Gauss map Q second fundamental form self-adjoint Q local coordinates
Q unpleasant expression Q invariants again… DDG Course SIGGRAPH 2005
9 Invariants
Gaussian and mean curvature Q determinant and trace only intrinsic extrinsic
Q eigen values and (ortho) vectors
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10 Curvatures
Integral representations Q smooth setting
Q on a mesh? Q hold on…
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11 Gaussian Curvature
On a mesh Q can’t take the limit… Q average does make sense only makes sense as an integral, NEVER pointwise
Discrete Gauss curvature at a vertex
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12 A Good Definition?
Gaussian curvature over a surface Q Gauss-Bonnet closed, oriented Q discrete
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13 Scalar Mean Curvature
Integral representation Q variation along a vector field
for constant V move out of integral
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14 Boundary Integrals
Vector area Q volume gradient: vector area
another normal
Q discrete version only makes sense as an integral, NEVER pointwise area weighted triangle normals DDG Course SIGGRAPH 2005
15 Boundary Integrals
Area gradient Q vector mean curvature
another normal
Q discrete version only makes sense as an integral, NEVER pointwise
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16 Boundary Integrals
Area gradient Q vector mean curvature
another normal
Q discrete version only makes sense as an integral, NEVER pointwise
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17 Laplace-(Beltrami)
Surface over tangent plane Q in eigen basis
principal curvature directions Q Laplace-Beltrami Laplace on …of the the surface surface
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18 Geometric Flow (Area)
Minimize area energy Q minimal surface
Hermann Schwarz, 1890 DiMarco, Physics, Montana
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19 Geometric Flow (Area)
Minimize area energy Q minimal surface
Hermann Schwarz, 1890 DiMarco, Physics, Montana
DDG Course SIGGRAPH 2005
20 Mean Curvature Flow
Laplace-Beltrami Q Dirichlet energy
Q on surface
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21 Mean Curvature Flow
Laplace-Beltrami Q Dirichlet energy
Q on surface
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22 Parameterizations
What is a parameterization? Q function from some region Ω ⊂ R2 to the embedded surface M⊂ R3 Image from Sander et al. 2002 al. et Sander from Image Q we go the other way around Q how to measure distortion?
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23 Measuring Distortion
Dirichlet energy of a map Q discrete harmonic
Q minimizer is discrete harmonic
angles in mesh texture coords. Q need to fix boundary DDG Course SIGGRAPH 2005
24 Harmonic Map
Properties of minimizer Q area minimization of triangle
conditions for u to be conformal DDG Course SIGGRAPH 2005
25 Discrete Conformal
Minimizer of conformal energy Q
Fixed boundary: Free boundary: Dirichlet → constant Neumann → match gradients
Dirichlet Neumann DDG Course SIGGRAPH 2005
26 Recap
Invariants as overarching theme Q shape does not depend on Euclidean motions Q metric and curvatures Q smooth continuous notions to discrete notions Q variational formulations Q careful: generally only as averages DDG Course SIGGRAPH 2005
27 Tools
Operators we have now Q volume gradient: notion of normal Q area gradient: notion of normal Q also: mean curvature & LB Harmonic Q smoothing, parameterization, editing (bi-Laplace-Beltrami) Flow G1 boundaries to Conformal make smooth bump
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28 Down the Line
Approach so far Q essentially linear: PL mesh… Q same equations can be derived with Q DEC: discrete exterior calculus Q abstract measure theory There is more Q some invariants are non-linear
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