cse291-c00 curves: continuous and discrete
TRANSCRIPT
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CSE291-C00
Curves: Continuous and Discrete
Instructor: Hao Su
Credit: Justin Solomon
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Defining “Curve”
A function?
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Subtlety
Not a curve
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Graphs of Smooth Functions
http://en.wikipedia.org/wiki/Singular_point_of_a_curve
“Cusp”
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Graphs of Smooth Functions
http://en.wikipedia.org/wiki/Singular_point_of_a_curve
“Cusp”
How to ensure the smoothness of a curve?
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Geometry of a Curve
A curve is a
set of pointswith certain properties.
It is not a function.
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Geometric Definition
Set of points that locally looks like a line.
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Differential Geometry Definition
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Parameterized Curve
Now this is OK!
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• Trace of parameterized curve
• Component functions
Some Vocabulary
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Change of Parameter
Geometric measurements should be
invariantto changes of parameter.
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Dependence of Velocity
On the board:Effect on velocity and acceleration.
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Arc Length
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Parameterization by Arc Length
Constant-speed parameterization
http://www.planetclegg.com/projects/WarpingTextToSplines.html
γ̄(s) = γ(t(s))
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Moving Frame in 2D
https://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas
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Philosophical Point
Differential geometry “should” be
coordinate-invariant.
Referring to x and y is a hack!(but sometimes convenient…)
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How do you characterize shape
without coordinates?
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Turtles All The Way Down
Use coordinates from the curve to express its shape!
On the board:
https://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas
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Radius of Curvature
https://www.quora.com/What-is-the-base-difference-between-radius-of-curvature-and-radius-of-gyration
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Fundamental theorem of the local theory of plane curves:
k(s) characterizes a planar curve up to rigid motion.
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Fundamental theorem of the local theory of plane curves:
k(s) characterizes a planar curve up to rigid motion.
Statement shorter than the name!
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• Binormal: • Curvature: In-plane motion• Torsion: Out-of-plane motion
Frenet Frame: Curves in
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Fundamental theorem of the local theory of space curves:
Curvature and torsion characterize a 3D curve up to
rigid motion.
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Aside: Generalized Frenet Frame
Gram-Schmidt on first n derivativesC. Jordan, 1874
Suspicion: Application to time series analysis? ML?
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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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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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Discrete Case
Homework
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For Small
Same behavior in the limithttp://en.wikipedia.org/wiki/Taylor_series
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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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Application
Structure Determination of Membrane Proteins Using Discrete Frenet Frameand Solid State NMR Restraints
Achuthan and QuineDiscrete Mathematics and its Applications, ed. M. Sethumadhavan (2006)
NMR scanner Kinked alpha helix
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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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Frenet Frame: Issue
?
http://upload.wikimedia.org/wikipedia/commons/6/6f/Frenet.png
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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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Curve-Angle Representation
Degrees of freedom for elastic energy:• Shape of curve• Twist angle
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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