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Duke CPS 296.1 - An Application to Curves

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142 VI PersistenceVI.3 An Application to CurvesIn this section, we use the stability of persistence to prove an inequality thatconnects the length and total curvature of two curves. We begin by recastingthe statement of stability in terms of continuous functions instead of filtrations.Sublevel sets. Let X be a topological space and g : X → R a continuousfunction. Given a threshold a ∈ R, the sublevel set consists of all pointsx ∈ X with function value less than or equal to a, Xa= g−1(∞, a]. Similarto the complexes in a filtration, the sublevel sets are nested and give rise tosequences of homology groups connected by maps induced by inclusion, onefor each dimension. Writing fa,bp: Hp(Xa) → Hp(Xb) for the map from thehomology group of the sublevel set for a to that for b, we call its image apersistent homology group. The corresponding persistent Betti number is βa,bp=rank im fa,bp. Furthermore, a ∈ R is a homological critical value if there is noε > 0 for which fa−ε,a+εpis an isomorphism for each dimension p. We assumethat g is tame, by which we mean that it has only finitely many homologicalcritical values and every sublevel set has only finite rank homology groups. Leta1< a2< . . . < anbe the homological critical values and b0< b1< . . . < bninterleaved values with bi−1< ai< bifor 1 ≤ i ≤ n. The 1-parameter familyof p-th homology groups can therefore be replaced by the finite sequence0 = Hb−1p→ Hb0p→ Hb1p→ . . . → Hbnp→ Hbn+1p= 0,where Hbp= Hp(Xb) and the groups at the two ends are added for convenience.Finally, we add a0= −∞ and an+1= ∞ to the list of critical values. For0 ≤ i < j ≤ n + 1, the multiplicity of the pair (ai, aj) is now defined asµi,jp= (βbi,bj−1p− βbi,bjp) − (βbi−1,bj−1p− βbi−1,bjp).To get the dimension p persistence diagram of g we draw each point (ai, aj)with multiplicity µi,jp. In contrast to the case of filtrations in which contiguouscomplexes differ by only one simplex, the multiplicities are no longer restrictedto 0 and 1 and there can be points at infinity. In particular for a bounded func-tion g the homology classes that get born but do not die correspond to pointsin the diagrams with ∞ as their second coordinate. With these definitions, wehave the following stability result which we state without proof.Stability Theorem for Functions. Let X be a triangulable topologicalspace, g, g0: X → R two tame functions, and p any dimension. Then thebottleneck distance between their diagrams satisfies dB(Dgmp(g), Dgmp(g0)) ≤kg − g0k∞.VI.3 An Appplication to Curves 143Closed curves. We consider a closed curve γ : S1→ R2, with or withoutself-intersections. Assuming γ is smooth, we have derivatives of all orders. Thespeed at a point γ(s) is the length of the velocity vector, k ˙γ(s)k. We can use itto compute the length as the integral over the curve,L(γ) =Zs∈S1k ˙γ(s)k ds.It is convenient to assume a constant speed parametrization, that is, % =k ˙γ(s)k =12πL(γ) for all s ∈ S1. With this assumption, the curvature at apoint γ(s) is the norm of the second derivative divided by the square of thespeed, κ(s) = k¨γ(s)k/%2. One over the curvature is the radius of the circle thatbest approximates the shape of the curve at the point γ(s). To interpret thisformula geometrically, we follow the velocity vector as we trace out the curve.Since its length is constant it sweeps out a circle of radius %, as illustratedin Figure VI.10. The curvature is the speed at which the unit tangent vector.γγγ.sss( )( )( )Figure VI.10: A curve with constant speed parametrization and its velocity vectorsweeping out a circle with radius equal to the speed.sweeps out the unit circle as we move the point with unit speed along the curve.This explains why we divide by the speed twice, first to compensate for thelength of the velocity vector and second to compensate for the actual speed.The total curvature is the distance traveled by the unit tangent vector,K(γ) =Zs∈S1%κ(s) ds.As an example consider the constant speed parametrization of the circle withradius r, γ(s) = rs. Writing a point of the unit circle in terms of its angle wegets =cos ϕsin ϕ, γ(s) =r cos ϕr sin ϕ, ˙γ(s) =−r sin ϕr cos ϕ.The constant speed is therefore % = r and the length is L(γ) =Rr ds = 2πr.The second derivative is ¨γ(s) = −γ(s) and the curvature is κ(s) =1rwhich is144 VI Persistenceindependent of the point on the circle. The total curvature is K(γ) =Rrrds =2π, which is independent of the radius. Indeed, the unit tangent vector travelsonce around the unit circle, no matter how small or how big the parametrizedcircle is.Integral geometry. The length and total curvature of a curve can also beexpressed in terms of integrals of elementary quantities. We begin with thelength. Take a unit length line segment in the plane. The lines that crossthe line segment at an angle ϕ form a strip of width sin ϕ. Integrating overall angles givesRπϕ=0sin ϕ dϕ = [− cos ϕ]π0= 2. In words, the integral of thenumber of intersections over all lines in the plane is twice the length of theline segment. Since we can approximate the curve by a polygon whose totallength approaches that of the curve, the same holds for the curve. To expressthis result, we write each line as the preimage of a linear function. Given adirection u ∈ S1let hu: R2→ R be defined hu(x) = hu, xi. The line withnormal direction u and offset z is h−1u(z). The intersections between γ and thisline corresponds to the preimage of the composition, h−1(z), where h = hu◦ γ.The length of the curve is therefore as given by the Cauchy-Crofton formula,L(γ) =14Zu∈S1Zz∈Rcard (h−1(z)) dz du.To get an alternative interpretation of the total curvature, we again consider adirection u ∈ S1and the height function in that direction, hu: S1→ R definedby hu(s) = hu, γ(s)i. For generic directions u, this height function has a finitenumber of minima and maxima, as illustrated in Figure VI.11. Recall that theuminminminminmaxmaxmaxmaxFigure VI.11: The vertical height function defined on the curve has four local minimawhich alternate with the four local maxima along the curve.total curvature is the length traveled by the unit tangent vector. Equivalently,it is the length traveled by the outward unit normal vector. The number ofmaxima of huis the number of times the unit normal passes over u ∈ S1andVI.3 An Appplication to Curves 145the


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Duke CPS 296.1 - An Application to Curves

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