curves: gauss map, turning number theorem, parallel transport · • the total number of times that...
TRANSCRIPT
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Curves: Gauss Map, Turning Number Theorem, Parallel Transport
Instructor: Hao Su
Credit: Justin Solomon
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What do these calculations look like in
software?
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Old-School Approach
Piecewise smooth approximations
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Question
What is the arc length of a cubic Bézier curve?
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Question
What is the arc length of a cubic Bézier curve?
Not known in closed form.
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Sad fact:Closed-form
expressions rarely exist. When they do exist, they
usually are messy.
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Only Approximations Anyway
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Equally Reasonable Approximation
Piecewise linear
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Big Problem
Boring differential structure
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Finite Difference Approach
THEOREM: As , [insert statement].
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Reality Check
THEOREM: As , [insert statement].
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Two Key Considerations
• Convergence to continuous theory
• Discrete behavior
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Goal
Examine discrete theories of differentiable curves.
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Goal
Examine discrete theories of differentiable curves.
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Gauss Map
http://mesh.brown.edu/3DPGP-2007/pdfs/sg06-course01.pdf
Normal map from curve to S1
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Signed Curvature on Plane Curves
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Winding Number• The total number of times that curve travels counterclockwise around the point.
• The winding number depends on the orientation of the curve, and is negative if the curve travels around the point clockwise.
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Winding Number
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Turning Numbers
http://mesh.brown.edu/3DPGP-2007/pdfs/sg06-course01.pdf
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Recovering Theta
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Turning Number Theorem
A “global” theorem!
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Discrete Gauss Map
http://mesh.brown.edu/3DPGP-2007/pdfs/sg06-course01.pdf
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Discrete Gauss Map
http://mesh.brown.edu/3DPGP-2007/pdfs/sg06-course01.pdf
Edges become points
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Discrete Gauss Map
http://mesh.brown.edu/3DPGP-2007/pdfs/sg06-course01.pdf
Vertices become arcs
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Key Observation
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What’s Going On?
Total change in curvature
( )
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What’s Going On?
Total change in curvature
( )
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What’s Going On?
Total change in curvature
( )
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Interesting Distinction
Same integrated curvature
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Interesting Distinction
Same integrated curvature
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What’s Going On?
Total change in curvature
( )
Integrated quantity
Dual cell
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Discrete Turning Angle Theorem
)(
)(
)(
)()(
)(
)( Preserved structure!
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Alternative Definition
Homework
decreases length the
fastest.
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Remaining Question
Does discrete curvature converge in limit?
Yes!
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Remaining Question
Does discrete curvature converge in limit?
Yes!
Questions:• Type of convergence?• Sampling?• Class of curves?
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Discrete Differential Geometry
• Different discrete behavior
• Same convergence
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Next
Curves in 3Dhttp://www.grasshopper3d.com/forum/topics/offseting-3d-curves-component
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Frenet Frame
http://upload.wikimedia.org/wikipedia/commons/6/6f/Frenet.png
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Potential Discretization
Discrete Frenet frame
Discrete frame introduced in:The resultant electric moment of complex molecules
Eyring, Physical Review, 39(4):746—748, 1932.
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Transfer Matrix
Discrete Frenet Frame, Inflection Point Solitons, and Curve Visualization with Applications to Folded Proteins
Hu, Lundgren, and NiemiPhysical Review E 83 (2011)
Discrete construction that works for fractal curves and converges in continuum limit.
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Segments Not Always Enough
http://www.cs.columbia.edu/cg/rods/
Discrete Elastic RodsBergou, Wardetzky, Robinson, Audoly, and Grinspun
SIGGRAPH 2008
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Simulation Goal
http://www.cs.columbia.edu/cg/rods/
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Adapted Framed Curve
Normal part encodes twisthttp://www.cs.columbia.edu/cg/rods/
Material frame
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Bending Energy
Punish turning the steering wheel
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Bending Energy
Punish turning the steering wheel
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Twisting Energy
Punish non-tangent change in material frame
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Twisting Energy
Punish non-tangent change in material frame
Swapping and does not affect !
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Which Basis to Use
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Frenet Frame: Issue
?
http://upload.wikimedia.org/wikipedia/commons/6/6f/Frenet.png
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Cross Product as Matrix Multiplication
[𝑎×] = − [𝑎×]𝑇
“skew-symmetric matrix”
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Darboux Vector of Frenet Frame
Ω
Ω
Ω
Ω
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Bishop Frame and its Darboux Vector
Most relaxed framehttp://www.cs.columbia.edu/cg/rods/
Darboux vector
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Bishop Frame
Most relaxed framehttp://www.cs.columbia.edu/cg/rods/
No twist(“parallel transport”)
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Curve-Angle Representation
Degrees of freedom for elastic energy:• Shape of curve• Twist angle
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Discrete Kirchoff Rods
Lower index: primal
Upper index: dual
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Discrete Kirchoff Rods
Tangent unambiguous on edge
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Discrete Kirchoff Rods
Integrated curvature
Yet another curvature!
Turning angle
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Discrete Kirchoff Rods
Darboux vector
Yet another curvature!Orthogonal to osculating plane, norm
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Bending Energy
Convert to pointwise and integrate
Can extend for natural bend
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Discrete Parallel Transport
• Map tangent to tangent• Preserve binormal• Orthogonal
http://www.cs.columbia.edu/cg/rods/
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Discrete Material Frame
http://www.cs.columbia.edu/cg/rods/
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Discrete Twisting Energy
Note can be arbitrary
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Simulation
\omit{physics}
Worth reading!
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Extension and Speedup
http://www.cs.columbia.edu/cg/threads/
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Extension and Speedup
“…the first numerical fluid-mechanical sewing machine.”
http://www.cs.columbia.edu/cg/threads/
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Morals
One curve,three curvatures.
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Morals
Easy theoretical object, hard to use.
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Morals
Proper frames and DOFs go a long way.
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Next
Surfaceshttp://graphics.stanford.edu/data/3Dscanrep/stanford-bunny-cebal-ssh.jpg
http://www.stat.washington.edu/wxs/images/BUNMID.gif