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A Geometric Invariant for the Study of Planar Curves and Its Application To A geometric invariant for the study of planar curves and its application to spiral tip meander. Copyright c 2016 Scott Hotton email: [email protected] Contents although in actual practice its often not feasi- ble to find an anti-derivative for x˙(τ)2 +y ˙(τ)2 1 Introduction 1 because of the square root. In any case local p curvature has been defined as κ(s)= dθ/ds. 2 The geometric invariant κ˘ 1 | | Although its usually not feasible to compute 3 Conservative examples 4 κ(s) it is usually not too difficult to compute x˙(t)¨y(t) y˙(t)¨x(t) 4 Dissipative examples 15 κ(s(t)) = − (x ˙(t)2 +y ˙(t)2)3/2 5 Conclusion 18 and this is often sufficient for many purposes. Its common practice to write κ(t) for κ(s(t)) and we 1 Introduction will use this convention here. Roughly speaking the total curvature of a 2 The geometric invariant κ˘ curve is how much θ changes over the whole | | curve. Historically total curvature has been de- Lets begin by reviewing some differential geom- fined as the integral of local curvature. This had etry of planar curves. We let t stand for time the drawback of making it seem that local cur- and we denote the position of the moving point vature needed to be defined in order for total at time t by (x(t),y(t))T . There are two impor- curvature to be defined. Fox and Milnor real- tant functions associated to twice differentiable ized however that total curvature is a meaningful planar curves, their speed and local curvature concept for all geometric curves [8, 15]. Moreover (more commonly referred to as just curvature). wherever local curvature is a meaningful concept We denote the speed by v(t) and the local cur- it can be defined in terms of total curvature so vature by κ(t). Also we will denote the direction total curvature is the more fundamental concept. of the velocity by θ S. Roughly speaking the Total curvature can still be computed from the ∈ arXiv:1602.07758v1 [q-bio.SC] 25 Feb 2016 local curvature tells us how fast θ is changing at integral of local curvature when local curvature a point of the curve. is defined but this is no longer regarded as a def- If the velocity is defined and never equal to inition of total curvature. This is now known (0, 0)T the curve is said to be immersed. If v(t) as the Fox-Milnor theorem and it is how we will ≡ 1 the curve is said to have unit speed. In theory compute total curvature here. immersed curves can be reparameterized to have Note that the Fox-Milnor theorem computes unit speed. This is done using the concept of arc total curvature as the integral of κ(s) with re- length. A closed form for arc length can some- spect to arc length not as the integral of κ(t) times be obtained from the integral with respect to time. By the change of variables theorem t t 2 2 s2 t2 t2 s(t)= v(τ) dτ = x˙(τ) +y ˙(τ) dτ κ(σ) dσ = κ(τ)v(τ) dτ = κ(τ) dτ 0 0 6 Z Z p Zs1 Zt1 Zt1 1 The integral for total curvature could be re- It follows thatϕ ˙(t) = κ(t)v(t) is a periodic formulated using θ˙(t) = κ(t)v(t). However if function with period T . The integral of a peri- we apply the fundamental theorem of calculus odic function is periodic if its average value over naively we might write one period is zero. And if we subtract the av- t2 erage value from a periodic function its integral θ˙(τ) dτ = θ(t ) θ(t ) will be periodic. So we set 2 − 1 Zt1 1 T but this is only the difference between the start- κ = v(τ)κ(τ) dτ ing direction and final direction of the velocity T 0 Z t and it overlooks the possibility that the veloc- ϕ(t) = κ(0)v(0) + κ(τ)v(τ) κ dτ ity may have undergone several complete turns − Z0 during the time interval [t ,t ]. To take this pos- 1 2 Thise allows us to write ϕ(t)= κt + ϕ(t) where sibility into consideration we define the function. ϕ(t) has period T . We let Rφ stands for a ro- t tation by φ radians. Even though ϕ(t) is not ϕ(t)= θ(0) + κ(τ)v(τ) dτ e periodice whenever κ = 0 it is the case that Z0 6 The set of all values for θ has the topology of Lemma 1. For all t R a circle whereas ϕ can take on any real number ∈ cos(ϕ(t)) cos(ϕ(t + T )) value since it is the integral of the real valued R = κ T sin(ϕ(t)) sin(ϕ(t + T )) function κ(t)v(t). We can recover θ from ϕ by taking its value modulo 2π. The rate of change of Proof. After making the substitution ϕ(t)= κt+ θ and ϕ are numerically equal, i.e. ϕ˙(t) = θ˙(t). ϕ(t) the proof is just a calculation which makes By the Fox-Milnor theorem the total curvature use of matrix multiplication, addition rules from of the curve from t = t1 to t = t2 is trigonometry,e and the fact that ϕ(t) has period t2 T . ϕ˙(τ) dτ = ϕ(t ) ϕ(t ) 2 − 1 e Zt1 The quantityκ ¯ T is the total curvature for The turning number is the total curvature di- the periodic arcs of the curve. We denote the vided by 2π. It measures how far the tangent turning number of a periodic arc byκ ˘. Note this vector has turned over the length of the curve. is independent of the choice of periodic arc. For The velocity of the curve can be expressed any t R 0 ∈ in terms of ϕ(t) as v(t) (cos(ϕ(t)), sin(ϕ(t)))T . t0+T T κ¯ T 1 Given the initial point of the curve, (x(0), y(0)) , κ˘ = = κ(τ)v(τ) dτ 2π 2π we can express the point at other times as Zt0 1 T x˙(τ)¨y(τ) y˙(τ)¨x(τ) x(t) x(0) t cos(ϕ(τ)) = − dτ (2.2) = + v(τ) dτ 2π x˙(τ)2 +y ˙(τ)2 y(t) y(0) sin(ϕ(τ)) Z0 Z0 (2.1) An advantage ofκ ˘ is that, unlike arc length, it doesn’t necessarily contain a radical under the We now assume that the speed and curvature integral which improves the prospects of finding are defined for all t R and that they are pe- ∈ an anti-derivative for use in the fundamental the- riodic functions with a common minimal period orem of calculus. T T > 0. This can occur by (x(t), y(t)) having Winfree coined the term “isogon contours” period T but this is not necessary. We call an in his study of spiral tip meander [19]. We say arc within such a curve whose domain is an in- here that an isogonal curve is a level curve ofκ ˘ terval of length T a periodic arc of the curve. We whetherκ ˘ is seen as a function in the state space show how to partition such curves into congruent or as a function in the parameter space. periodic arcs below. 2 An even congruence is a congruence of the Using the change of variables theorem with η = Euclidean plane which preserves the orientation τ + T gives of the plane. An odd congruence reverses the ori- t cos(ϕ(τ + T )) entation of the plane. Total curvature is invari- v(τ + T ) dτ = ant under even congruences and turned into its sin(ϕ(τ + T )) Z0 negative by odd congruences. Thus the quantity t+T cos(ϕ(η)) T cos(ϕ(η)) v(η) dη v(η) dη κ˘ is invariant under all congruences. It gives us sin(ϕ(η)) − sin(ϕ(η)) a| geometric| property of the curve. In particular Z0 Z0 we can express some of the curve’s symmetries in terms ofκ ˘. Let Therefore x(t) x(t) x(0) x(t + T ) x(T ) = R2πκ˘ R2πκ˘ = Gκ,T˘ y(t) y(t) − y(0) y(t + T ) − y(T ) x(T ) x(0) x(t) R + R (2.3) which can be rearranged to give the theorem. y(T ) − 2πκ˘ y(0) 2πκ˘ y(t) Whenκ ˘ Z the rotation R reduces to the ∈ 2πκ˘ identity map and is a translation by the vec- Gκ,T˘ tor (x(T ) x(0), y(T ) y(0))T . Otherwise The turning number − − Gκ,T˘ is a rotation by 2πκ˘ modulo 2π radians about for each periodic arc the point is -2/3 x¯ 1 x(T ) x(0) = Rπ(1/2 κ˘) − y¯ 2 sin(πκ˘) − y(T ) y(0) − Theorem 2. For all t R ∈ x(t) x(t + T ) = Gκ,T˘ y(t) y(t + T ) Proof. Moving (x(0), y(0))T from the right hand Figure 1: A curve withκ ˘ = 2/3. The curve is side of (2.1) to the left hand side and applying − partitioned into three periodic arcs. The tangent the rotation R2πκ˘ to both sides gives vector rotates by minus two thirds of a turn from x(t) x(0) R the starting point of a periodic arc to its final 2πκ˘ y(t) − y(0) point.
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