A Geometric Invariant for the Study of Planar Curves and Its Application To

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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)ÿ(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. The tangent vector rotates by minus two t cos(ϕ(τ)) whole turns along the entire closed curve so its = R2πκ˘ v(τ) dτ 0 sin(ϕ(τ)) Whitney turning number is 2. The curve has Z − t cos(ϕ(τ)) no reflectional symmetry and its full symmetry = v(τ)R2πκ˘ dτ 0 sin(ϕ(τ)) group is generated by 2/3,T . Z G− By Lemma 1 We can arbitrarily pick any point on the curve, t cos(ϕ(τ)) T v(τ)R dτ (x(t1), y(t1)) , and apply κ,T˘ to it to get the 2πκ˘ sin(ϕ(τ)) TG Z0 point (x(t1 + T ),y(t1 + T )) . The portion of the t cos(ϕ(τ + T )) curve between these two points is a periodic arc. = v(τ) dτ sin(ϕ(τ + T )) We can apply to this periodic arc to get an Z0 Gκ,T˘ adjacent periodic arc and so on. We can apply and since v(t) has period T the inverse of κ,T˘ to get the rest of the curve on t G T cos(ϕ(τ + T )) the other side of (x(t1),y(t1)) . In this way we v(τ) dτ sin(ϕ(τ + T )) can partition the curve into periodic arcs start- Z0 t cos(ϕ(τ + T )) ing from any point. A simple example is shown = v(τ + T ) dτ sin(ϕ(τ + T )) in figure 1. Z0 3 The quantityκ ˘ is like the Whitney turning Tracing point number [17] for closed curves except the Whit- ney turning number is a topological invariant whereasκ ˘ is a geometric invariant. Also the value ofκ ˘ can be any real number whereas the Epicycle Whitney turning number must be an integer. Whenκ ˘ is a non-integral rational number p/q (p, q coprime) there is a close relationship between Deferent the two quantities. A periodic arc will return to itself after being rotated q times by . This Gp/q,T implies the image of the curve is closed and that the Whitney turning number of its image is p. Conversely given a closed curve with Whitney turning number p and rotational symmetry by p/q of a turn the curve can be partitioned into q arcs with total turning number p/q. This is illustrated in figure 1. Figure 2: Epicyclic motion is the spin of the epicycle as its center revolves about the center 3 Conservative examples of the deferent. It can be expressed as the sum of two vectors which turn with fixed angular ve- Example 1 - Epicyclic motion locities. Epicyclic motion was an ancient Greek model for our solar system 1. It was a fair approxima- of the epicycle. We denote these two vectors at tion for the motion of the planets as seen in the time t = 0 as (x ,y )T and (x ,y )T respectively. Earth’s rest frame but of course it has long since 1 1 2 2 Figure 2 presents a simple time parametrization been superseded by Heliocentric models. for curves traced out by epicyclic motion. Epicyclic motion is formed by combining two Epicyclic motion can be generated by pro- rotary motions. A point on a circle, called the jecting the orbits for a pair of uncoupled har- deferent, revolves with a constant angular veloc- monic oscillators. Harmonic oscillators are con- ity ω1. At each moment in time the revolving servative systems. In this case the dynamics can point is the center for a another circle called the be obtained from the Hamiltonian epicycle which spins about its center with con- 1 H(x ,y , x ,y )= ω x2 + y2 + ω x2 + y2 stant angular velocity ω2. A chosen point on the 1 1 2 2 −2 1 1 1 2 2 2 epicycle, that we will call the tracing point, stood for the location of a planet (see figure 2). The Because of the symmetry of this Hamiltonian orbit of a planet was represented by the curve it is arbitrary which pair of variables, x1, x2 or generated by the tracing point. y1, y2, are regarded as the positions and which At any moment in time the location of the are regarded as the momenta. The choice only tracing point can be expressed as the sum of two affects the direction the system moves along the vectors: a position vector for the location of the orbits. We choose x1, x2 to be the position vari- center of the epicycle relative to the center of ables and y1, y2 to be conjugate momenta. Each T the deferent and a position vector for the loca- (xj,yj) pair moves with angular frequency ωj tion of the tracing point relative to the center along a circle in the state space. T Projecting (x1,y1,x2,y2) to the pair of posi- 1Prominent figures in the development of this model T tion variables, (x1,x2) , gives us Lissajous curves. were Apollonius, Hipparchus, and Ptolemy. In Ptolemy’s We will consider Lissajous curves only briefly. version the center was offset from the Earth but this did not change the shape of the curve in the Earth’s rest This will occur in the following example on the frame. spherical pendulum. To obtain epicyclic motion

4 from uncouple harmonic oscillators we instead They can be constructed as roulettes, i.e. by T T project (x1,y1,x2,y2) to (x1 + y1, x2 + y2) . one curve rolling without slipping along another T We set rj = (xj,yj) for j = 1, 2. For all curve. One of the simplest non-trivial roulettes t R we have is generated by a point on a disk rolling without ∈ slipping along a line. These are called trochoids. r r (x(t),y(t))T r + r | 1 − 2|≤ ≤ 1 2 The curves generated by epicyclic motion can be generated by a circular disc rolling without We call r r the minimum radius and r + r | 1 − 2| 1 2 slipping along another circular disc. This con- the maximum radius. The curve attains its T struction method was perhaps originally conceived minimum radius when Rω1t((x1,y1) ) and T of by Dürer in 1525 and then again by the as- Rω2t((x2,y2) ) point in opposite directions and tronomer Rømer in 1624. These curves have it attains its maximum radius when they point been studied by many mathematicians since. in the same direction. By suitably shifting time For some purposes it is useful to allow the we can suppose, without loss of generality, that tracing point to be outside of the rolling disc. at t = 0 the vectors point in the same direction. We can treat the union of the rolling disc and This simplifies the time parameterization of the the tracing point as a single rigid body even curve to: when their union does not form a connected set. x(t) r r This is done by applying the same motion of = R 1 + R 2 (3.1) y(t) ω1t 0 ω2t 0 the rolling disc to the tracing point regardless of where the tracing point happens to be. And T If ω1 = ω2 then the vectors Rω1t(r1, 0) , since we are allowing the tracing point to be out- T Rω2t(r2, 0) will continue to point in the same side of the rolling disc, we can dispense with the direction and the tracing point will travel in a disc’s interior in the definitions and just work circle. Also if ω1 = 0 or ω2 = 0 the tracing with a pair of circles, one fixed and one rolling point will travel in a circle so in this section we along other. assume that ω ω (ω ω ) = 0. Furthermore, The pair of circles are required to intersect 1 2 2 − 1 6 for the purpose of comparing these curves to the in exactly one point. When they intersect in ex- curves generate by spiral tip meander in section actly one point they have the same tangent line 4, we will assume ω1 > 0 which is to say the at the contact point, hence the circles are said to epicycle revolves anticlockwise. be tangent to each other. If neither circle is in- T Since ω1 = ω2 the vectors Rω1t(r1, 0) , side the other then they are said to be externally T 6 Rω2t(r2, 0) will alternately point in the same tangent. Otherwise they are said to be inter- and opposite directions. The condition that they nally tangent. If the fixed and rolling circles are point in the same or opposite direction is equiv- externally tangent then the curve generated by alent to the tracing point is called an epitrochoid. If the

(cos(ω1t), sin(ω1t), 0) (cos(ω2t), sin(ω2t), 0) = fixed circle is outside of the rolling circle then || × || the curve is called a hypotrochoid. If the fixed sin((ω2 ω1)t)=0 − circle is inside of the rolling circle then the curve Thus configurations for the deferent, epicycle, is called a peritrochoid. and tracing point which are congruent to the ini- If the tracing point is inside of the rolling cir- tial condition occur with a periodicity of T = cle then the hypotrochoid, epitrochoid, or per- 2π/ ω ω . It can be checked that this is the itrochoid is said to be curtate. If the tracing | 2 − 1| common minimal period for v(t) and κ(t). So we point is outside of the rolling circle then the hy- can apply the theory from the previous section potrochoid, epitrochoid, or peritrochoid is said to epicyclic motion. to be prolate 2. If the tracing point is on the

The curves generated by the tracing point are 2 Some authors reverse the meaning of curtate and pro- not properly called epicycles. These curves have late, e.g. [4] names based on a diﬀerent construction method.

5 rolling circle then the hypotrochoid, epitrochoid, epicycle are generally not the same as the fixed or peritrochoid is called a hypocycloid, epicycloid, and rolling circles in the roulette construction. or pericycloid respectively However the fixed circle can be concentric with the deferent and at any moment in time the rolling circle can be concentric with the epicycle. Only the radii may need to be different. To determine the correct radii for the fixed and rolling circles we make use of the no slip condition. The rolling circle may not slip as it goes around the fixed circle. This means that the instantaneous velocity of the contact point is the zero vector (0, 0)T . For epicyclic motion there is exactly one point, at any given moment in time, that is instantaneously at rest so this must be where the contact point for the fixed and rolling circles are at that moment. The contact point of any pair of tangent circles is collinear with their centers and by convention their centers are on the x-axis at t = 0 so at this time the contact point is on the x-axis as well. We let a denote the x-coordinate of the contact point at time t = 0. So a is the radius | | of the fixed circle. T At t = 0 the vector (r1, 0) is rotating with T angular velocity ω1 about (0, 0) while the vector T T (a, 0) (r1, 0) is rotating with angular velocity − T ω2 about (r1, 0) . Taking the derivative with Figure 3: Three configurations for the fixed and respect to time, evaluating at t = 0, and setting rolling circles. We assume the rolling circles re- the result equal to the zero vector gives volve anti-clockwise as indicated by the arrows 0 r1 a r1 at their centers. This causes the rolling circle = ω1Rπ/2 + ω2Rπ/2 0 0 0 − 0 to turn clockwise for hypotrochoids and anticlockwise for epitrochoids and peritrochoids as which has the unique solution a = (1 ω /ω )r − 1 2 1 indicated by the arrows on the rolling circles. (since ω = 0). To get the radius of the rolling 2 6 wheel we set b = a r , i.e. the directed distance − 1 Its useful to have one term which encom- from the center of the rolling circle to the contact passes hypotrochoids, epitrochoids, and peritro- point. The radius of the rolling circle is b . | | choids. Morely recognized this in 1894 and pro- The center of the fixed circle is at (0, 0)T and posed using just “trochoids” [12] even though at t = 0 the center of the rolling circle is at T this term is often restricted to the case where a (r1, 0) . If ω1/ω2 < 0 then r1 < a, the contact circle rolls along a line. More recently the term point is on the right hand side of both circles, “centered trochoid” has been proposed [3]. We and the fixed circle is outside of the rolling cir- shall use central trochoid and use the term cen- cle. So the curve is a hypotrochoid (see figure tral cycloid to denote a hypocycloid, epicycloid, 3). If 0 < ω1/ω2 < 1 then 0 < a < r1, the or pericycloid. The central cycloids are those contact point is between the circles, and the cir- central trochoids that have cusps. cles are externally tangent. So the curve is an Although the curves generated by epicyclic epitrochoid. If 1 < ω1/ω2 then a < 0, the con- motion are central trochoids the deferent and tact point is on the left hand side of both circles,

6 and the fixed circle is inside of the rolling circle. cloid, and when ρ > 1 the central trochoid is So the curve is a peritrochoid. prolate. From these facts the usual parameterizations Another geometrical invariant that we will for the central trochoids in terms of the radii a , use is the ratio of angular velocities χ = ω /ω | | 1 2 b and the angle φ = ω t can easily be derived. It (recall ω = 0). We call χ the turning ratio. | | 1 2 6 is convenient to work with (3.1) because each of It can also be expressed as χ = (b/a)/(b/a 1). − the epicyclic parameters occurs just once in the The quantity b/a is often called the wheel ratio. | | expression. It is helpful to keep in mind, though, We will see that for curtate central trochoids the the different ways that central trochoids can be wheel ratio equalsκ ˘ (for ω1 > 0). constructed, e.g. either by epicyclic motion or Even with ρ > 0 if χ = 0, 1 then the curve as a roulette. that is traced out is a circle which we do not wish to include as a type of central trochoid. When χ< 0 the central trochoid is a hypocycloid, when 0 <χ< 1 the central trochoid is an epitrochoid, and when 1 <χ the central trochoid is a peritrochoid (see figure 4). We set

= (χ, ρ) χ< 0, ρ> 0 H { | } = (χ, ρ) 0 <χ< 1, ρ> 0 E { | } = (χ, ρ) 1 <χ, ρ> 0 P { | } and = . T H∪ E ∪ P In addition to the central cycloids there are three other cases of central trochoids worth dis- Figure 4: The spaces of hypotrochoids, , of H tinguishing. These cases also form curves in epitrochoids, , and of peritrochoids, . Along T E P as shown in ﬁgure 4. The ﬁrst of these cases is with the four distinguished cases, central cy- given by the vertical line χ = 1. The equation cloids, ellipses, billiard like, and rhodoneas. − for the hypotrochoid reduces to

We are primarily concerned with the shape of x(t) (r + r ) cos(ω t) = 1 2 1 the central trochoids rather than their position, y(t) (r1 r2) sin(ω1t) orientation, or size. So we will determine the − space of similarity classes of central trochoids. which determines an ellipse with semi-major axis For this purpose we introduce two geometrically r1 + r2 and semi-minor axis r1 r2 . | − | invariant parameters for the central trochoids. We call the cases given by the diagonal lines One commonly used geometric invariant for ρ = χ “billiard like”. These curves have long | | describing the shape of a central trochoid is ρ = arcs with low curvature alternating with short ω /ω r /r . This is often called the arm ratio. arcs with high curvature. They are fairly well | 2 1| 2 1 This is because the line segment connecting the approximated by the paths made by a friction- center of the rolling circle to the tracing point less billiard ball rolling on a circular table which is often called the arm and ρ is the ratio of the travels along straight line segments and bounce arm’s length to the radius of the rolling circle, at the table’s edge. r / b = ρ. When ρ = 0 the resulting curve is The remaining distinguished case of central 2 | | just a single point if ω1 = 0 and a circle other- trochoids corresponds to the central trochoid pass- wise. We do not wish to regard these as special ing through its own center of symmetry. The cases of central trochoids so we require ρ > 0. minimum radius of a central trochoid is zero if When 0 <ρ< 1 the central trochoid is curtate, and only if the deferent and epicycle have the when ρ = 1 the central trochoid is a central cy- same size. When r1 = r2 the distance of the

7 forth along a line segment. For the Tusi couple κ˘ hypo- epi- peri- the tangent vector is undefined at the end points 1 1 1 of the line segment, as with the cusps of central prolate χ 1 1 χ χ 1 − − − cycloids. The line segment can be regarded as a χ χ χ degenerate ellipse. The line segment is literally curtate χ 1 1 χ χ 1 − − − a billiard curve for a round table so we can say Table 1: The value ofκ ˘ for central trochoids it is a billiard like curve that passes through its when the rolling wheel revolves in the anticlock- center of symmetry. wise direction, ω1 > 0. For ω1 < 0 take the The Tusi couple is the only instance in which negative of each table entry. the curvature of a central trochoid is zero at any point. Hence central trochoids do not have inflection points. tracing point from the center of the curve, as a The value ofκ ˘ for central trochoids can be function of time, is computed from equation (2.2). It almost reduces to an uncomplicated function of χ except that it 2r1 cos(((ω1 ω2)/2) t) − depends on the signs of ω , ρ 1, and χ(χ 1). 1 − − which is essentially the defining condition for The formulas forκ ˘ are shown in table 1. For any 3 “rhodonea” curves . The condition r1 = r2 is real number except zero there is a central tro- equivalent to ρ χ = 1 which specifies a pair of choid whose value forκ ˘ is the given real number. | | hyperbolic arcs in (see figure 4). These hyper- Combining χ = (b/a)/(b/a 1) withκ ˘ = T − bolic arcs form the subspace of rhodonea curves. χ/(χ 1) givesκ ˘ = b/a. So we see from ta- − Below the hyperbolic arcs the deferent is larger ble 1 thatκ ˘ = b/a for curtate hypotrochoids than the epicycle while above the hyperbolic arcs and peritrochoids. For curtate epitrochoidsκ ˘ = the deferent is smaller than the epicycle. b/a. The signs of a, b are the same for hy- − potrochoids and peritrochoids and opposite for curtate epitrochoids. Therefore the total turning number for a periodic arc of a curtate central trochoid is the same as the wheel ratio, i.e. κ˘ = b/a . | | The isogonal curves in are vertical line seg- T ments which span the heights of the curtate and prolate regions. When the line of central cycloids is crossed the value ofκ ˘ changes by 1 (see figure ± 5). The central trochoids undergo a cusp transition. In this transition a loop is either added Figure 5: The isogonal curves in are vertical or removed from each periodic arc of the central T line segments. They are labeled with their value trochoid. forκ ˘. The value ofκ ˘ changes by 1 when the A sufficient, but not necessary, condition for ± line of cycloids is crossed. The region in is two central trochoids to be similar is for them E shaded to facilitate comparison to figure 4. to have the same values for χ and ρ. We will obtain the spaces of similarity classes of central In the sets of hypocycloids, ellipses, bil- trochoids by quotienting each of the spaces , H H liard like curves, and rhodoneas all intersect at , and . E P one point (χ, ρ) = ( 1, 1). This particular spe- In models for the planetary motion the epicy- − 4 cial case is known as the “Tusi couple” . For cle was generally much smaller than the defer- the Tusi couple the tracing point goes back and ent. Actually though, because vector addition 3Studied by Guido Grandi around 1723 is commutative, it doesn’t matter which of the 4Studied by Nasir al-Din al-Tusi around 1247 two circles we take to be the deferent and epicy-

8 Figure 7: The space, , of similarity classes of Figure 6: The space, , of similarity classes of E H Epitrochoids. The negative real axis and the Hypotrochoids. The origin corresponds to the origin make up the complement of in C. The E Tusi couple. The four compass directions corre- other three compass directions correspond to the spond to the four distinguished cases. The isogo- three distinguished cases of Epitrochoids. The nal curves are shown in gray and they are labeled isogonal curves are shown in gray and they are with their κ˘ values. The value of κ˘ jumps as an labeled with their κ˘ values. The value of κ˘ | | | | | | | | isogonal curve passes through the line of hypocy- jumps as the isogonal curve passes through the cloids. line of epicycloids. cle. If we let the center of the deferent revolve merely describe how the curves can be constructed about the center of the epicycle with angular ve- geometrically. Many geometric figures can be locity ω while the deferent spins with angular 2 constructed in more than one way. This termi- velocity ω then the exact same curve can be 1 nology can lead to some confusion particularly traced out. However when we look for fixed and since peritrochoids have been regarded as epitro- rolling circles to generate the central trochoid we choids by some authors while other authors have get a different location for the point at instanta- regarded them as hypotrochoids. Willson gives neous rest and thus different radii for the fixed an overview on this terminology in the appendix and rolling circles. This fact is known as David to his 1898 book [18]. Bernoulli’s “dual generation theorem”. Every central trochoid, except for central cy- A consequence of the dual generation theo- cloids, can be constructed as a curtate central rem is that the ordered pair (1/χ, 1/ρ) deter- trochoid and as a prolate central trochoid. The mines the same similarity class of central tro- value of κ˘ equals the wheel ratio in the cur- choids as (χ, ρ). A curtate hypotrochoid is geo- | | tate method of construction. We can think of metrically similar to a prolate hypotrochoid and κ˘ as a generalization of the wheel ratio to other visa verse. A prolate epitrochoid is similar to a types of curves with periodically varying curva- curtate peritrochoid and visa versa. A curtate ture. Althoughκ ˘ can equal 0 for some curves epitrochoid is similar to a prolate peritrochoid with periodically varying curvature but not for and visa versa. central trochoids. The terms “hypotrochoid”, “epitrochoid”, “per- Suppose we have two central trochoids with itrochoid”, “curtate”, and “prolate” do not des- parameters ω , ω , r , r and ω , ω , r , r . A ignate geometric similarity classes of curves but 1 2 1 2 1′ 2′ 1′ 2′

9 necessary condition for two central trochoids to is onto the complex plane C minus the non- be similar is for the ratio of their minimum radius positive real axis. It is two to one everywhere to their maximum radius be the same, i.e. in . The point with the same image as E ∪ P (χ, ρ) is (1/χ, 1/ρ) so we can identify with the r1 r2 r′ r′ E | − | = | 1 − 2| slitted complex plane (see figure 7). r + r r + r 1 2 1′ 2′ The value of χ is positive for an epitrochoid Since the radii are positive this equation is equiv- or peritrochoid (for ω1 > 0) so regardless of alent to the equation whether it is prolate or curtateκ> ˘ 0. The projection from to maps pairs of isogonal curves r r r r E E 1′ 1 1′ 2 = 0 with the sameκ ˘ value to the same semiparabolic r − r r − r 2′ 2 2′ 1 arc in . To obviate the issue of how the Epitro- E choids are parameterized we associate the value If r1′ /r2′ = r1/r2 then its necessary for ω1′ /ω2′ = ω /ω and if r /r = r /r then its necessary for of κ˘ to each semiparabolic arc. If the difference 1 2 1′ 2′ 2 1 | | ω /ω = ω /ω . Therefore the only points in in κ˘ between a semiparabolic arc in the lower 1′ 2′ 2 1 T | | that correspond to the same similarity class as half-plane of and a semiparabolic arc in the E (χ, ρ) is (1/χ, 1/ρ). upper half-plane of is 1 then their union is the E We denote the space of similarity classes of entire parabola except for its vertex on the line hypotrochoids by . To denote the members of Epicycloids. H of we use the term Hypotrochoid with the first Example 2 - The spherical pendulum H letter capitalized. The word “hypotrochoid” with The spherical pendulum is an idealized mechan- all lower case letters describes how the curve was ical system. It is comprised of a weightless in- constructed. For (χ, ρ) the function extensible rod, essentially a line segment with ∈ H length ℓ. One end of the rod, the pivot, is mo- (χ, ρ) (log( χ)+ i log(ρ))2 7→ − tionless for all time. The other end, the bob, is onto the complex plane, C, and it is two to is the location of a point particle with mass m. one everywhere except at the Tusi couple. The The bob is constrained to move on a sphere of point with the same image as (χ, ρ) is (1/χ, 1/ρ) radius ℓ centered at the pivot while subjected to so we can identify with C (see figure 6). a uniform gravitational field with strength g. H The value of χ is negative for hypotrochoids We take the pivot to be the origin of a Carte- so regardless of whether it is prolate or curtate sian coordinate system for the spherical pendu- 1 < κ<˘ 1. The projection from to maps lum. We take the direction of the gravitational − H H the isogonal curves withκ ˘ = 1/2 to the line field to be the negative direction of the z-axis. ± of ellipses. Otherwise it maps pairs of isogo- The z-axis is also called the pendulum’s axis. nal curves with opposite sign to the same semi- The plane through the pivot and orthogonal to parabolic arc in . We can associate the value of the pendulum’s axis is the support plane. The H κ˘ to each semiparabolic arc. The semiparabolic support plane contains the (x,y)-axes of the co- | | arcs whose values for κ˘ sum to 1 form the en- ordinate system (see figure 8). Because the spher- | | tire parabola except for its vertex on the line of ical pendulum is symmetrical about its axis the Hypocycloids. orientation of the (x,y)-axes is completely arbi- We denote the space of similarity classes of trary. epitrochoids by . For members of we use We also use spherical coordinates to specify E E the term Epitrochoid with the first letter capi- the position of the bob. Because the letters θ, ϕ talized. The words “epitrochoid” and “peritro- are used through out this article to describe the choid” with all lower case letters describes how direction of the velocity along planar curves we the curve was constructed. For (χ, ρ) let ϑ stand for the polar angle of the spherical ∈ E ∪ P the function coordinate system and ψ stand for the azimuthal angle (see figure 8). (χ, ρ) (log(χ)+ i log(ρ))2 7→

10 to the support plane is quasiperiodic. It will be shown that the pendulum’s period is the com- z mon minimal period of v(t), κ(t) for the (x,y)- curves so we can apply the theory from section 2. The kinetic energy of the bob is 1 1 m(x ˙ 2 +y ˙2 +z ˙2)= mℓ2(ϑ˙2 + sin2(ϑ)ψ˙2) 2 2 and the potential energy is U = mgz = mgℓ cos(ϑ). y Since U is independent of ϑ˙ and ψ˙ the conjugate x momenta are ∂ 1 P = mℓ2(ϑ˙2 + sin2(ϑ)ψ˙2)= mℓ2ϑ˙ ϑ ˙ 2 Figure 8: Cartesian coordinates, (x,y,z), ∂ϑ ∂ 1 and spherical coordinates, (ϑ, ψ) for the bob J = mℓ2(ϑ˙2 + sin2(ϑ)ψ˙2)= mℓ2 sin2(ϑ)ψ˙ ∂ψ˙ 2 of a spherical pendulum. They are related by (x,y,z)= ℓ (sin(ϑ) cos(ψ), sin(ϑ) sin(ψ), cos(ϑ)). These are the horizontal and vertical compo- nents of the bob’s total angular momentum respectively. The state of the spherical pendulum The theory presented in section 2 can be gen- is completely specified by the canonical variables eralized to curves on a sphere using the concept (ϑ, Pϑ,ψ,J). The Hamiltonian is of geodesic curvature but to avoid unnecessary 2 1 2 J H(ϑ, Pϑ,ψ,J)= P + +mgℓ cos(ϑ) complications in this example we will study the 2mℓ2 ϑ sin2(ϑ) path of the bob from a bird’s eye view. More pre- (3.2) cisely stated we orthogonally project the bob’s path on the sphere into the support plane. We The fact that H is independent of ψ shows treat the spherical pendulum as merely a mech- us that the value of J is constant. Although anism for generating planar curves which are the the state space is four dimensional the dynam- objects of our study here. ics can be reduced to two dimensions because We present a short review of the spherical H, J are constant. Moreover the reduced sys- pendulum’s dynamics. The bob moves in three tem has a nondimensionalized form. Physically, dimensional space but it is subject to a single the parameters, m, g, ℓ, are limited to positive holonomic constraint so it has two degrees of values and varying them does not produce any freedom. Two independent integrals of motion qualitative changes in behavior so long as they are the total energy or Hamiltonian, H and the remain positive. To obtain the reduced system vertical component of its angular momentum, J. we nondimensionalized the constants of motion, The system is fully integrable. Its orbits are ei- we define a dimensionless potential energy and ther quasiperiodic, periodic, or fixed points. its dimensionless rate of change, and we define a The projection of the bob’s position to the dimensionless time: pendulum’s axis is always periodic. When the H J bob’s z-coordinate has a minimal period it is (h, j) = , mgℓ mℓ√gℓ sometimes referred to as the pendulum’s period T even if the pendulum is behaving quasiperiodi- T U ℓ cally. So long as J = 0 the bob’s (x,y) coordi- (u, w) = , u˙ 6 mgℓ sg ! nates will be rotated about the pendulum’s axis by a nonzero amount during the pendulum’s pe- t = g/ℓ t riod. Typically the angle for this rotation is ir- The equations of motionp which can be obtained rational so the overall motion and its projection from the Hamiltonian (3.2) can be used to show

11 that (u, w)T satisﬁes the diﬀerential equation

du/dt w = (3.3) dw/dt 3u2 2hu 1 − − This is known as the reduced system for the spherical pendulum. Its is not a straight forward initial value problem. First of all the initial value for u must be in the interval [ 1, 1]. Secondly it − follows from (3.2) that j 1 w2 + j2 h = + u (3.4) 2 1 u2 − Although (u, w)T varies with time the value of h depends on (u, w)T in such a way that it does not change with time. The value of h can be determined by (3.4) from the initial value for (u, w)T and the constant value for j. Once the value of h has been determined from the initial conditions it can be treated as a fixed parameter in (3.3). The (x,y)-curves can be obtained from a solution for u by using a single quadrature. Figure 9: The energy-momentum space, P, for √g/ℓ t dτ the spherical pendulum is the light gray region ψ(t) = ψ(0) + j 2 0 1 u(τ) bounded by the black j curves. The isogo- Z − ± max x(t) cos(ψ(t)) nal curves, shown in dark gray, are labeled with = ℓ 1 u(t)2 (3.5) y(t) − sin(ψ(t)) their values forκ ˘. They radiate from (1, 0). The p dark gray oval, 27j2 = 2h(9 4h2), is where in- − The maximum potential energy is attained flection points partition the corresponding (x,y)- when the bob is directly above the pivot and the curves into periodic arcs. Outside of the oval the minimum potential energy is attained when the (x,y)-curves do not have inflection points. Insets bob is directly below the pivot. If the bob has no show (x,y)-curves in gray each with a periodic kinetic energy when its directly above or below arc shown in black. Dotted arrows point to the the pivot then it will remain where it is. These corresponding (h, j) values. two states are the fixed points of the Hamiltonian system (3.2). If h = 1 then the bob must be − is ((h √h2 + 3)/3, 0)T . The maximum value motionless directly below the pivot. This is the − for j is minimum possible value for h. There is no limit to how fast the bob can move so h has no upper 2 2 3/2 3 jmax = 3 ((h + 3) h + 9h) bound. 9 − q For each h 1 there is a finite range of and the minimum value is jmax. The graph of ≥ − − values for j. The extreme values for j can be jmax is an increasing, concave down, curve with obtained by rearranging (3.4) to a single end point as shown in figure 9. The set of all possible values for (h, j) is the energy- 1 1 j2 = w2 (h u)(1 u2) (3.6) momentum space for the spherical pendulum, − 2 2 − − − P = (h, j) : h 1, j j and differentiating the right hand side with re- { ≥− | |≤ max} spect to u and w while treating h as a fixed pa- For j = 0 the angular velocity ψ˙ is always rameter. There is one critical point for j which zero and the pendulum moves within a vertical

12 plane. In this case it is often called a planar pendulum even though there are no physical forces constraining it within a plane. If (h, j) = ( 1, 0) − the (x,y)-curve is just a point. If (h, j) = (1, 0) the bob can be directly above the pivot or it can move asymptotically towards that position. For each h such that h > 1 and h = 1 all of the − 6 (x,y)-curves corresponding to (h, 0) are line segments with the same length. The right hand side of (3.6) can be taken as a Hamiltonian function for the reduced system (3.3). Thus the critical point of j corresponds to a fixed point of the reduced system. So the extreme values for j are attained when w = 0, i.e. the potential energy is constant. In these cases the height of the bob does not change. This is often referred to as a conical pendulum because the rod sweeps out a cone. The bob rotates along a horizontal circle with the sign of j determining the direction of rotation. For each h> 1 all of − the (x,y)-curves determined by (h, j ) are ± max the same circle. Figure 10: The projection of the invariant tori For each (h, j) in corresponding to given (h, j) values into the coordinate space for (x, y, w). The location of the R = (h, j) : h> 1, 0 < j < jmax P { − | | } (h, j) values in is shown in figure 9. Each torus shows the projection of a rotationally sym- the subset of the spherical pendulum’s state space metric quasiperiodic orbit in the torus. which is mapped to (h, j) is a torus5 [2]. The action of the group of rotations about the pendulum’s axis can be extended to the whole state the period of the reduced system in dimension- space of the pendulum and for each (h, j) R less time t. T depends on (h, j). In physical time ∈ the corresponding torus is invariant under this t the period is ℓ/g T . We obtain the velocity action. Rotations about the pendulum’s axis of the (x,y)-curves from (3.5) and (3.6). p have no effect on w and these tori can be pro- v = gℓ (2h 2u w2) jected into a three dimensional space while pre- − − serving their symmetry by using the coordinates The period of 2hp 2u w2 is the same as the (x, y, w) (see figure 10). For each (h, j) R − − ∈ period of the reduced system and since the ex- all of the corresponding (x,y)-curves are rotated pression under the radical is positive for all time copies of each other. The orbits typically wind for (h, j) R the period of v is the same as the quasiperiodically around the torus but they can ∈ period of the reduced system. We also obtain be periodic. In either case the curvature of the the curvature from (3.5) and (3.6). (x,y)-curve varies periodically. The value ofκ ˘ is defined for the corresponding (x,y)-curve if and x˙ y¨ y˙x¨ 3 2h 3u only if (h, j) R. κ = −3 = j g ℓ −3 ∈ v v Since the shape of the (x,y)-curves is com- p pletely determined by (h, j) R we can think of which has the same period. From equation (2.2) ∈ κ˘ as a function on the space R. We let T denote and the change of variables theorem we get an expression forκ ˘ entirely in terms of the reduced 5Cushman’s R space is a little larger than this.

13 system. κ˘ can be within any tiny distance below 1 but | | it can not be 1 or more. For h > 1 the value j T 2h 3u κ˘ = − dτ (3.7) ofκ ˘ appears to converge to 1 as j approaches 0 2π 2h 2u w2 Z0 from above while it appears to converge to 1 as − − − A few numerically computed isogonal curves in j approaches 0 from below. As (h, j) crosses the R are shown in figure 9. The isogonal curves horizontal axis of P the (x,y)-curves transition radiate from the point (h, j) = (1, 0). by collapsing to a line segment. It is common to approximate the behavior of In the value ofκ ˘ converges to 1/2 as (χ, ρ) H the spherical pendulum when (h, j) is near the approaches the Tusi couple, ( 1, 1), from below − vertex ( 1, 0) P by linearizing the system and it converges to 1/2 as (χ, ρ) approaches − ∈ − about its fixed point for (h, j) = ( 1, 0). In this the Tusi couple from above. As (χ, ρ) crosses − case we might expect that the (x,y)-curves for the Tusi couple the (x,y)-curves transition by the spherical pendulum could be well approxi- collapsing to a line segment. mated by Lissajous curves since we are project- In P there is a half-line for which the cor- ing the state variables to the position variables responding (x,y)-curves are line segments while (x,y). However because the spherical pendulum in there is a single point for which the (x,y)- H is symmetrical about its axis the two frequencies curves is a line segment. On the other hand in of the linearized system are equal and therefore, P the isogonal curves radiant from a single point regardless of the initial conditions, the only type while in all of the isogonal curves pass through H of Lissajous curves generated by projecting the a half-line. state variables of the linearized system to the po- There are some important differences between sition variables are ellipses or line segments. Fur- in P and . The value ofκ ˘ for billiard like Hy- H thermore the eigenvalues of the linearized system potrochoids must be in the interval [ 1/2, 1/2] − are purely imaginary so the Hartman-Grobman so the (x,y)-curves generated by the spherical theorem does not apply [5], i.e. there need not be pendulum do not resemble billiard like Hypotro- a neighborhood in which the spherical pendulum choids. They also do not resemble rhodonea curves is equivalent to its linearization. The spherical since they only pass through their center of sym- pendulum is in fact highly nonlinear [11]. No metry when they collapse to line segments. And matter how close (h, j) R is to ( 1, 0) the unlike Hypotrochoids an (x,y)-curve for the spher- ∈ − (x,y)-curves resemble Hypotrochoids more than ical pendulum can have inflection points. This they do Lissajous curves (see insets in figure 9), happens when (h, j) is inside the oval shown in which is not too surprising given the geometry figure 9. of the spherical pendulum. The value ofκ ˘ is associated with the intrin- It is interesting to compare the spaces P and sic precession of the spherical pendulum. The R (figure 9) with the spaces (figure 5) and motion of the projected image of the bob in the H H (figure 6). For these spaces the reflection about support plane can be thought of as a compound the horizontal axis maps isogonal curves to isog- motion of a point around an ellipse with the ro- onal curves and the range of observed values for tation of the ellipse about its center. The intrin- κ˘ is in the open interval ( 1, 1). Also in these sic precession of the spherical pendulum is the − spaces the range of observed values for κ˘ below rotation of the ellipse. | | the horizontal axis is the open interval (1/2, 1). It should be briefly pointed out that the in- Numerical analysis indicates that in R the trinsic precession of a spherical pendulum is dis- value of κ˘ can be within any tiny distance above tinct from Foucault precession. The difference | | 1/2 but it can not be 1/2 or less. For h ( 1, 1) was recognized by Foucault himself. As is well ∈ − the value ofκ ˘ appears to converge to 1/2 as j ap- known, Foucault designed and built a spherical proaches 0 from above while it appears to con- pendulum in 1851 to measure the rotation of the verge to 1/2 as j approaches 0 from below. Nu- Earth [9]. Foucault’s pendulum was designed to − merical analysis also indicates that the value of be set librating within a vertical plane. Since the

14 pendulum’s support is rotating with the Earth odic curvature. These are the Shenoy-Rutenberg the plane of libration appears to rotate relative model for the paths taken by the bacterium Lis- to the ground below it. This motion is called teria monocytogenes in eukaryotic cells [6] and Foucault precession. the Barkley-Kevrekidis model for the meander Foucault found that it can be difficult to start of spiral waves in excitable media such as the a spherical pendulum with sufficiently little an- BZ reaction [1]. gular momentum so that it will appear to oscil- Example 3 - Actin based motility late within a vertical plane. Even a small amount L. monocytogenes transport themselves in eu- of angular momentum led to an intrinsic preces- karyotic cells by catalyzing the polymerization sion comparable to the Foucault precession. To of the cytoskeletal protein actin. This method overcome this apparent “instability” he designed of propulsion can result in a bacterium following his pendulum with very large m and ℓ. a complicated path within the cytosol at a fairly If j is small enough the periodic arcs of the | | constant speed. The curvature of the paths tends (x,y)-curve generated by the spherical pendulum to vary periodically with time so the curvature can be fairly well approximated by the periodic and speed have a common minimal period and arcs of an ellipse (see top left inset in figure 9). we can apply the theory from section 2. The approximating ellipse turns in the same di- Recall from section 2 that the velocity’s ori- rection as the bob so the total curvature of a entation is ϕ(t) = κ t + ϕ(t) where ϕ(t) is pe- periodic arc of the (x,y)-curve of the spherical riodic. In the Shenoy-Rutenberg model ϕ(t) = pendulum is the sum of the total curvature of (Ω/ω0) sin(ω0t) where ω0 ise the angulare frequency a periodic arc of the approximating ellipse with of the bacterium’s spin about its long axise and Ω the intrinsic precession of the approximating el- is a monotonic function of the distance of the ef- lipse. For any nondegenerate ellipse the value of fective propulsive force from the long axis. In the the total curvature of a periodic arc isκ ˘ = 1/2 ± model the speed of the bacterium is the constant (recall figure 6). The amount of intrinsic preces- v0. The common minimal period of the curva- sion that occurs during the pendulum’s period ture and speed is T = 2π/ω0 and so κ = ω0κ˘. is The points of maximal curvature occur for t κ˘ 1/2 forκ> ˘ 1/2 ∈ − (2π/ω )Z and the points of minimal curvature κ˘ + 1/2 forκ< ˘ 1/2 0 − occur for t (π/ω ) + (2π/ω )Z. ∈ 0 0 If j is large the periodic arcs of the (x,y)-curve Since (Ω/ω ) sin(ω t) is an odd function the | | 0 0 are not well approximated by periodic arcs of full image of the curve has reflectional symme- an ellipse (see bottom left inset in figure 9). In try. Joining an arc over a half-period with its these cases it is not very helpful to think of the reflected image gives a periodic arc. For integral (x,y)-curve as being generated by a precessing κ˘ the rest of the curve can be obtained by trans- ellipse. lating the periodic arc. For non-integralκ ˘ the The value ofκ ˘ is defined for all (h, j) R rest of the curve can be obtained by rotating the ∈ regardless of how poorly the periodic arcs of an periodic arc about (x, y)T . (x,y)-curve can be approximated by the periodic When the initial direction is horizontal, i.e. arcs of an ellipse. The value ofκ ˘ tells us how ϕ(0) = 0, and the starting point is at the origin, far the (x,y)-curve turns during the pendulum’s i.e. (x(0),y(0))T = (0, 0)T , the time parameter- period as well as providing us with information ization for the curve is about the symmetry of the (x,y)-curve. x(t) t cos(ω κτ˘ + (Ω/ω ) sin(ω τ)) = v 0 0 0 dτ y(t) 0 sin(ω κτ˘ + (Ω/ω ) sin(ω τ)) 0 0 0 0 Z (4.1) 4 Dissipative examples There is no closed form for this integral in terms of elementary functions but we can obtain a closed In this section we consider two related models for form that accurately approximates it by borrow- natural systems that generate curves with peri- ing a technique from civil engineering known as

15 fected by this approximation since only the ϕ(t) term in ϕ(t) is altered and the reﬂectional symmetry is unaﬀected since ϕ(t) remains an even e function. Because of the symmetry the error varies periodically and so remains bounded. The e quantity

16 arcsin(cos(ω t/2)) arcsin(sin(ω t/2)) sin(ω t) π2 0 0 − 0

o is never more than 3.21 at any time. Figure 12 shows an example of a spiral easement approximation for the parametrized curve in equation (4.1) We let ( (t), (t))T denote the time param- X Y eterization for the approximating curve to (4.1). On the interval [0,π/ω0] the curvature can be written as the linear polynomial 2β(α βt) where − πω κ˘ + 4Ω 2 ω Ω α = 0 β = 0 2πβv π v 0 r 0

Integrating over a sub-interval [0,t] [0,π/ω0] gives us a closed form time parameterization⊆ for an arc of the ( (t), (t))-curve, X Y

(t) v0 C(α βt)+ C(α) X = R 2 − − (4.2) Figure 11: Generating a spiral easement approx- (t) β (α ) S(α βt) S(α) imation for an (x,y)-curve. Above: A clothoid. Y − − It is symmetrical under a half turn and the cen- where C(t), S(t) are the Fresnel trigonometric ter is its unique inflection point. Below: The functions. approximating ( , )-curve. The two arcs high- X Y lighted in black are geometrically similar to each other. The ( , )-curve can be obtained by suc- X Y cessively reflecting its black arc about the five mirror lines shown in figure 12. spiral easement. This technique varies the curvature of roads and train tracks in a piecewise linear fashion. For the Shenoy-Rutenberg model we approximate the curvature with the triangu- lar wave form ω κ˘ Ω 8 κ(t) 0 + arcsin(cos(ω t)) ≈ v v π2 0 0 0 This gives a piecewise quadratic approximation for ϕ(t): Figure 12: An (x,y)-curve with (˘κ, Ω/ω0) = (4/5, 1) along with its approximating ( , )- Ω 16 X Y 2 arcsin(cos(ω0 t/2)) arcsin(sin(ω0 t/2)) ω0e π curve translated to have the same center. The curves’ five mirror lines are also displayed. The rotational symmetry of the curve is unaf-

16 The planar curve t (C(t),S(t))T is known curvature pass through the center together and 7→ as a clothoid6. It is shown at the top of figure then proceed outward. Afterwards the points of 11. It has unit speed so t is the arc length from minimal curvature pass back through the center it center, (0, 0)T to (C(t),S(t))T . The curva- again. The process is reminiscent of a loom ex- ture at (C(t),S(t))T is 2t so every possible curva- cept the curve becomes wound up around four ture occurs at exactly one point of the clothoid. points (whenκ ˘ = 1/2) instead of being woven. Since the curvature is monotonic the clothoid does not intersect itself and since the curvature is unbounded in the positive and negative directions the clothoid spirals around two points. The right hand side of equation (4.2) is the application of a odd similarity transformation to an arc of the clothoid. This approximation technique amounts to taking the extremal curvatures of the (x,y)T -curve, determining the two points on the clothoid where these extremal curvatures occur, and applying an odd similarity to the arc in the clothoid connecting the points of extremal curvature as shown in figure 11. The rest of the ( , )-curve is obtained by the action of the X Y symmetry group of the ( , )-curve. The ap- X Y proximation can be further refined by translating the ( , )-curve so that it has the same center X Y as the (x,y)-curve (see figure 12). The value ofκ ˘ determines the rotational sym- Figure 13: ( , )-curves with fixedκ ˘ = 1/2 and X Y metry of the curve. The value of Ω/ω0 deter- increasing Ω/ω0. Each row shows a pair of points mines the length of the clothoid arc used in ap- with extremal curvature passing through each proximating the curve. The effect of Ω/ω0 for other at the center thereby introducing a pair of fixedκ ˘ is perhaps best illustrated in theκ ˘ = 1/2 crossing points which persist as Ω/ω0 continues case (see figure 13) because there are fewer self- to increase. intersections to deal with. More generally, forκ ˘ / Z, the points of max- As Ω/ω varies the points of minimal curva- 0 imal curvature coincide∈ with the center if and ture oscillate in unison along mirror lines and only if ( (0), (0))T = ( , )T while the points the points of maximal curvature oscillate in uni- of minimalX curvatureY coincideX Y with the center if T T son along mirror lines. For small Ω/ω0 the curve and only if ( (π/ω0), (π/ω0)) = ( , ) . If has an oval shape with the points of minimal the points of minimalX curvatureY coincideX withY the curvature closer to the center than the points center then of maximal curvature. As Ω/ω0 increases the cos(α2) (C(γ) C(α)) + sin(α2) (S(γ) S(α))=0 points of minimal curvature move toward the − − (4.3) center and inflection points appear. Next the where γ = α βπ/ω . This gives us a condition − 0 points of minimal curvature pass through the which must hold betweenκ ˘ and Ω/ω0. The same center together and then move outward. Eventu- condition holds if the points of maximal curva- ally the points of minimal curvature go far from ture coincide with the center exceptκ ˘ is replaced the center while the points of maximal curva- with κ˘. These two conditions determine two − ture go near the center. The points of maximal sets of curves in the (˘κ, Ω/ω0) parameter plane 6 which are shown in figure 14. It is also known as Euler’s spiral and as Cornu’s spiral. These two sets of curves only intersect at integer values ofκ ˘. It turns out in these cases that

17 the congruence reduces to the identity map, From equation (2.2) the total curvature per Gκ,T˘ that the image of the ( , )-curve is closed, and periodic arc is X Y thatκ ˘ is its turning number. √7 ν √14π/7 κ˘ w(τ) dτ ≈ π Z0

where ν = γ0/√28. The isogonal curves forκ ˘ = 1, 2, 3, 4 are shown in ﬁgure 15 where µ = (α + − 2 5)/5.

Figure 14: The parameter plane for equation Figure 15: (Above) Blow up of a region in fig- (4.2) and the curves defined by (4.3). The ure 14. (Below) The (ν,µ) parameter space solid black curves correspond to ( , )-curves for Barkley’s ODE (compare to figure 3 in [1]). X Y whose points of minimal curvature coincide with Curves in the two parameter spaces are depicted the center. The dotted black curves correspond in the same manner as in figure 14. ( , )-curves whose points of maximal curva- X Y ture coincide with the center. The solid black and dotted black curves only intersect on the ver- 5 Conclusion tical gray lines which correspond to integer values forκ ˘. The gray diagonal lines correspond to the appearance of inflection points in the ( , )- References X Y curves. [1] Barkley, D. (1994). Euclidean Symme- try and the Dynamics of Rotating Spiral Example 4 - Spiral tip meander Waves. Phy. Rev. Lett., 72(1), pp. 164– Barkley’s model [1] for the spiral wave tip me- 167. ander can be written as the ordinary differential equation [2] Cushman, R. (1983). Geometry of the En- ergy Momentum Mapping of the Spherical x˙ v cos(ϕ) Pendulum. Centrum voor Wiskunde en In- y˙ v sin(ϕ)     formatica Newsletter, 1, 4–18. ϕ˙ = γ0 w v˙  v ( 1/4 + (10/3)v2 + α w2 v4) [3] Ferréol, R. (2015). Trocho¨ıde A` Centre.    − 2 −  w˙   w ( 1+ v2 w2)  http://www.mathcurve.com/courbes2d    − −      /trochoid/trochoidacentre.shtml Here we have replaced the variable name ‘s’ in (accessed July 2015). Barkley’s equations (3) and (4) with the variable name ‘v’ in keeping with the convention in this [4] Ganguli, Surendramohan (1926). The The- article that s stands for arc length and v stands ory of Plane Curves Volume II, Second edi- for speed. tion. University of Calcutta press.

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