Turning NumberNumber of orbits in Gauß image

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Different homotopy classes in image

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 ordered on curve, ordered straight edges

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LengthSum of edge lengths

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LengthSmooth curve limit of inscribed polygon lengths limit of inscribed polygon lengths

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Total Signed CurvatureSum of turning angles

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Discrete Gauß MapEdges map to points, vertices map to arcsto arcs

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Discrete Gauß MapTurning number well-defined for discrete curvesdiscrete curves

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Turning Number TheoremClosed curve the total signed curvature is an the total signed curvature is an

integer multiple of 2. proof: sum of exterior angles

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Structure-PreservationArbitrary discrete curve total signed curvature obeys total signed curvature obeys

discrete turning number theorem even on a coarse mesh can be crucial

d di h li i

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depending on the application

ConvergenceConsider refinement sequence length of inscribed polygon to length of inscribed polygon to

length of smooth curve discrete measure approaches

continuous analogue which refinement sequence?

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which refinement sequence? depends on discrete operator pathological sequences may exist

Total Signed CurvatureSum of turning angles

Recall:

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Another DefinitionCurvature normal

signed unit

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gcurvature(scalar)

normal(vector)

Curvature normalGradient of length define discrete curvature 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

ConvergenceStructure-

preservationpreservation

In the limit of a refinement sequence, discrete

measures of length and curvature agree with

continuous measures.

For an arbitrary (even coarse) discrete curve, the

discrete measure of curvature obeys the

discrete turning number th

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theorem.