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Turning Number
Number of orbits in Gauß image
Different homotopy classes in image
CS177 (2012) - Discrete Differential Geometry
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Turning Number Thm.
For a closed curve
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Discrete Setting
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Inscribed Polygon:
Finite number of vertices
on curve,
curve ordered
 straight edges

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Length
Sum of edge lengths
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Length
Smooth curve

limit of inscribed polygon lengths
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Total Signed Curvature
Sum of turning angles
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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
discrete curves
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Turning Number Theorem
Closed curve
the total signed curvature is an
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 on a coarse mesh
 can be crucial


d
depending
di on the
h application
li i
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Convergence
Consider refinement sequence
length of inscribed polygon to
length of smooth curve
 discrete measure approaches
continuous analogue
 which refinement sequence?



depends on discrete operator
pathological sequences may exist
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Recall:
Total Signed Curvature
Sum of turning angles
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Another Definition
Curvature normal
signed
g
curvature
(scalar)
unit
normal
(vector)
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Curvature normal
Gradient of length

define discrete curvature
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Gradient of Length
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Gradient of Length
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Gradient of Length
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Gradient of Length
+
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Gradient of Length
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Gradient of Length
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Gradient of Length
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Moral of the Story
Structurepreservation
For an arbitrary (even
coarse) discrete curve, the
discrete measure of
curvature obeys the
discrete turning number
th
theorem.
Convergence
In the limit of a refinement
sequence, discrete
measures of length and
curvature agree with
continuous measures.
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