a first look at ddg: discrete curves
TRANSCRIPT
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A First Look at DDG:
Discrete Curves
Eitan Grinspun, Columbia University
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Part I: plane curves
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Secant
A line through two points on the curve.
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Secant
A line through two points on the curve.
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Tangent, the first approximant
The limiting secant as the two pointscome together.
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Circle of curvature
Consider the circle passing through threepoints on the curve…
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Circle of curvature
…the limiting circle as three points cometogether.
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Radius of curvature,
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Radius of curvature,
Curvature
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Signed curvature
Sense of traversal along curve.
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Gauß map,
Point on curve maps to point on unit circle.
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Shape operator
Change in normal as we slide along curve
• describes curvature
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Turning number,
Number of orbits in Gaussian image.
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Turning number theorem
For a closed curve,the integral of curvature isan integer multiple of 2 .
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Part II: discrete plane curves
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Inscribed polygon,
Finite number of verticeseach lying on the curve,connected by straight edges.
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The length of a discrete curve
Sum of edge lengths.
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The length of a continuous curve
Length of longest of all inscribed polygons.
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The length of a continuous curve
…or take limit over refinement sequence.
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Total signed curvature
Sum of turningangles.
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Discrete Gauß Map
Edges map to points, vertices map to arcs.
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Discrete Gauß Map
Turning number well-defined for discretecurves.
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Discrete Turning Number Theorem
For a closed curve,the total signed curvature isan integer multiple of 2 .• proof: sum of exterior angles
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Structure-preservation
Arbitrary discrete curve• total signed curvature obeys
discrete turning number theorem• even coarse mesh• which continuous theorems to preserve?
• that depends on the application• fast-forward to last lecture:
– Euclidian motions? triangle mesh is fine
– Conformal maps? use circle-based mesh
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Structure-preservation
Arbitrary discrete curve• total signed curvature obeys
discrete turning number theorem• even coarse mesh• which continuous theorems to preserve?
• that depends on the application• fast-forward to last lecture:
– Euclidian motions? triangle mesh is fine
– Conformal maps? use circle-based mesh
discrete analogueof continuous theorem
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Convergence
Consider refinement sequence• length of inscribed polygon approaches
length of smooth curve• in general, discrete measure approaches
continuous analogue• which refinement sequence?
• depends on discrete operator• pathological sequences may exist
• in what sense does the operator converge?(point-wise, L2; linear,quadratic)
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Total signed curvature
Sum of turningangles.
Recall:
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Other definitions for curvature
“Curvature normal”
signedcurvature(scalar)
unitnormal(vector)
[Kobbelt and Schröder]
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Curvature normal = length gradient
Use this to define discrete curvature!
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Curvature normal = length gradient
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Curvature normal = length gradient
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Curvature normal = length gradient
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Curvature normal = length gradient
+
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Curvature normal = length gradient
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Curvature normal = length gradient
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Curvature normal = length gradient
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Recap
ConvergenceStructure-preservation
In the limit of arefinement sequence,discrete measures oflength and curvatureagree with continuousmeasures.
For an arbitrary (evencoarse) discrete curve,the discrete measure ofcurvature obeys thediscrete turning numbertheorem.