A geometric invariant for the study of planar curves and its application to spiral tip meander
aa r X i v : . [ q - b i o . S C ] F e b A geometric invariant for the study of planar curves and itsapplication to spiral tip meander.
Copyright c (cid:13) [email protected]
Contents | ˘ κ |
13 Conservative examples 44 Dissipative examples 155 Conclusion 18 | ˘ κ | Lets begin by reviewing some differential geom-etry of planar curves. We let t stand for timeand we denote the position of the moving pointat time t by ( x ( t ) , y ( t )) T . There are two impor-tant functions associated to twice differentiableplanar curves, their speed and local curvature(more commonly referred to as just curvature).We denote the speed by v ( t ) and the local cur-vature by κ ( t ). Also we will denote the directionof the velocity by θ ∈ S . Roughly speaking thelocal curvature tells us how fast θ is changing ata point of the curve.If the velocity is defined and never equal to(0 , T the curve is said to be immersed . If v ( t ) ≡ unit speed . In theoryimmersed curves can be reparameterized to haveunit speed. This is done using the concept of arclength. A closed form for arc length can some-times be obtained from the integral s ( t ) = Z t v ( τ ) dτ = Z t p ˙ x ( τ ) + ˙ y ( τ ) dτ although in actual practice its often not feasi-ble to find an anti-derivative for p ˙ x ( τ ) + ˙ y ( τ ) because of the square root. In any case localcurvature has been defined as κ ( s ) = dθ/ds .Although its usually not feasible to compute κ ( s ) it is usually not too difficult to compute κ ( s ( t )) = ˙ x ( t )¨ y ( t ) − ˙ y ( t )¨ x ( t )( ˙ x ( t ) + ˙ y ( t ) ) / and this is often sufficient for many purposes. Itscommon practice to write κ ( t ) for κ ( s ( t )) and wewill use this convention here.Roughly speaking the total curvature of acurve is how much θ changes over the wholecurve. Historically total curvature has been de-fined as the integral of local curvature. This hadthe drawback of making it seem that local cur-vature needed to be defined in order for totalcurvature to be defined. Fox and Milnor real-ized however that total curvature is a meaningfulconcept for all geometric curves [8, 15]. Moreoverwherever local curvature is a meaningful conceptit can be defined in terms of total curvature sototal curvature is the more fundamental concept.Total curvature can still be computed from theintegral of local curvature when local curvatureis defined but this is no longer regarded as a def-inition of total curvature. This is now knownas the Fox-Milnor theorem and it is how we willcompute total curvature here.Note that the Fox-Milnor theorem computestotal curvature as the integral of κ ( s ) with re-spect to arc length not as the integral of κ ( t )with respect to time. By the change of variablestheorem Z s s κ ( σ ) dσ = Z t t κ ( τ ) v ( τ ) dτ = Z t t κ ( τ ) dτ θ ( t ) = κ ( t ) v ( t ). However ifwe apply the fundamental theorem of calculusnaively we might write Z t t ˙ θ ( τ ) dτ = θ ( t ) − θ ( t )but this is only the difference between the start-ing direction and final direction of the velocityand it overlooks the possibility that the veloc-ity may have undergone several complete turnsduring the time interval [ t , t ]. To take this pos-sibility into consideration we define the function. ϕ ( t ) = θ (0) + Z t κ ( τ ) v ( τ ) dτ The set of all values for θ has the topology ofa circle whereas ϕ can take on any real numbervalue since it is the integral of the real valuedfunction κ ( t ) v ( t ). We can recover θ from ϕ bytaking its value modulo 2 π . The rate of change of θ and ϕ are numerically equal, i.e. ˙ ϕ ( t ) = ˙ θ ( t ).By the Fox-Milnor theorem the total curvatureof the curve from t = t to t = t is Z t t ˙ ϕ ( τ ) dτ = ϕ ( t ) − ϕ ( t )The turning number is the total curvature di-vided by 2 π . It measures how far the tangentvector has turned over the length of the curve.The velocity of the curve can be expressedin terms of ϕ ( t ) as v ( t ) (cos( ϕ ( t )) , sin( ϕ ( t ))) T .Given the initial point of the curve, ( x (0) , y (0)) T ,we can express the point at other times as (cid:18) x ( t ) y ( t ) (cid:19) = (cid:18) x (0) y (0) (cid:19) + Z t v ( τ ) (cid:18) cos( ϕ ( τ ))sin( ϕ ( τ )) (cid:19) dτ (2.1)We now assume that the speed and curvatureare defined for all t ∈ R and that they are pe-riodic functions with a common minimal period T >
0. This can occur by ( x ( t ) , y ( t )) T havingperiod T but this is not necessary. We call anarc within such a curve whose domain is an in-terval of length T a periodic arc of the curve. Weshow how to partition such curves into congruentperiodic arcs below. It follows that ˙ ϕ ( t ) = κ ( t ) v ( t ) is a periodicfunction with period T . The integral of a peri-odic function is periodic if its average value overone period is zero. And if we subtract the av-erage value from a periodic function its integralwill be periodic. So we set κ = 1 T Z T v ( τ ) κ ( τ ) dτ e ϕ ( t ) = κ (0) v (0) + Z t κ ( τ ) v ( τ ) − κ dτ This allows us to write ϕ ( t ) = κ t + e ϕ ( t ) where e ϕ ( t ) has period T . We let R φ stands for a ro-tation by φ radians. Even though ϕ ( t ) is notperiodic whenever κ = 0 it is the case that Lemma 1.
For all t ∈ R R κ T (cid:18) cos( ϕ ( t ))sin( ϕ ( t )) (cid:19) = (cid:18) cos( ϕ ( t + T ))sin( ϕ ( t + T )) (cid:19) Proof.
After making the substitution ϕ ( t ) = κt + e ϕ ( t ) the proof is just a calculation which makesuse of matrix multiplication, addition rules fromtrigonometry, and the fact that e ϕ ( t ) has period T . The quantity ¯ κ T is the total curvature forthe periodic arcs of the curve. We denote theturning number of a periodic arc by ˘ κ . Note thisis independent of the choice of periodic arc. Forany t ∈ R ˘ κ = ¯ κ T π = 12 π Z t + Tt κ ( τ ) v ( τ ) dτ = 12 π Z T ˙ x ( τ )¨ y ( τ ) − ˙ y ( τ )¨ x ( τ )˙ x ( τ ) + ˙ y ( τ ) dτ (2.2)An advantage of ˘ κ is that, unlike arc length, itdoesn’t necessarily contain a radical under theintegral which improves the prospects of findingan anti-derivative for use in the fundamental the-orem of calculus.Winfree coined the term “isogon contours”in his study of spiral tip meander [19]. We sayhere that an isogonal curve is a level curve of ˘ κ whether ˘ κ is seen as a function in the state spaceor as a function in the parameter space.2n even congruence is a congruence of theEuclidean plane which preserves the orientationof the plane. An odd congruence reverses the ori-entation of the plane. Total curvature is invari-ant under even congruences and turned into itsnegative by odd congruences. Thus the quantity | ˘ κ | is invariant under all congruences. It gives usa geometric property of the curve. In particularwe can express some of the curve’s symmetriesin terms of ˘ κ . Let G ˘ κ,T (cid:18) x ( t ) y ( t ) (cid:19) = (cid:18)(cid:18) x ( T ) y ( T ) (cid:19) − R π ˘ κ (cid:18) x (0) y (0) (cid:19)(cid:19) + R π ˘ κ (cid:18) x ( t ) y ( t ) (cid:19) (2.3) When ˘ κ ∈ Z the rotation R π ˘ κ reduces to theidentity map and G ˘ κ,T is a translation by the vec-tor ( x ( T ) − x (0) , y ( T ) − y (0)) T . Otherwise G ˘ κ,T is a rotation by 2 π ˘ κ modulo 2 π radians aboutthe point (cid:18) ¯ x ¯ y (cid:19) = 12 sin( π ˘ κ ) R π (1 / − ˘ κ ) (cid:18) x ( T ) − x (0) y ( T ) − y (0) (cid:19) Theorem 2.
For all t ∈ R G ˘ κ,T (cid:18) x ( t ) y ( t ) (cid:19) = (cid:18) x ( t + T ) y ( t + T ) (cid:19) Proof.
Moving ( x (0) , y (0)) T from the right handside of (2.1) to the left hand side and applyingthe rotation R π ˘ κ to both sides gives R π ˘ κ (cid:18)(cid:18) x ( t ) y ( t ) (cid:19) − (cid:18) x (0) y (0) (cid:19)(cid:19) = R π ˘ κ Z t v ( τ ) (cid:18) cos( ϕ ( τ ))sin( ϕ ( τ )) (cid:19) dτ = Z t v ( τ ) R π ˘ κ (cid:18) cos( ϕ ( τ ))sin( ϕ ( τ )) (cid:19) dτ By Lemma 1 Z t v ( τ ) R π ˘ κ (cid:18) cos( ϕ ( τ ))sin( ϕ ( τ )) (cid:19) dτ = Z t v ( τ ) (cid:18) cos( ϕ ( τ + T ))sin( ϕ ( τ + T )) (cid:19) dτ and since v ( t ) has period T Z t v ( τ ) (cid:18) cos( ϕ ( τ + T ))sin( ϕ ( τ + T )) (cid:19) dτ = Z t v ( τ + T ) (cid:18) cos( ϕ ( τ + T ))sin( ϕ ( τ + T )) (cid:19) dτ Using the change of variables theorem with η = τ + T gives Z t v ( τ + T ) (cid:18) cos( ϕ ( τ + T ))sin( ϕ ( τ + T )) (cid:19) dτ = Z t + T v ( η ) (cid:18) cos( ϕ ( η ))sin( ϕ ( η )) (cid:19) dη − Z T v ( η ) (cid:18) cos( ϕ ( η ))sin( ϕ ( η )) (cid:19) dη Therefore R π ˘ κ (cid:18) x ( t ) y ( t ) (cid:19) − R π ˘ κ (cid:18) x (0) y (0) (cid:19) = (cid:18) x ( t + T ) y ( t + T ) (cid:19) − (cid:18) x ( T ) y ( T ) (cid:19) which can be rearranged to give the theorem. The turning numberfor each periodic arcis -2/3
Figure 1: A curve with ˘ κ = − /
3. The curve ispartitioned into three periodic arcs. The tangentvector rotates by minus two thirds of a turn fromthe starting point of a periodic arc to its finalpoint. The tangent vector rotates by minus twowhole turns along the entire closed curve so itsWhitney turning number is −
2. The curve hasno reflectional symmetry and its full symmetrygroup is generated by G − / ,T .We can arbitrarily pick any point on the curve,( x ( t ) , y ( t )) T , and apply G ˘ κ,T to it to get thepoint ( x ( t + T ) , y ( t + T )) T . The portion of thecurve between these two points is a periodic arc.We can apply G ˘ κ,T to this periodic arc to get anadjacent periodic arc and so on. We can applythe inverse of G ˘ κ,T to get the rest of the curve onthe other side of ( x ( t ) , y ( t )) T . In this way wecan partition the curve into periodic arcs start-ing from any point. A simple example is shownin figure 1.3he quantity ˘ κ is like the Whitney turningnumber [17] for closed curves except the Whit-ney turning number is a topological invariantwhereas ˘ κ is a geometric invariant. Also thevalue of ˘ κ can be any real number whereas theWhitney turning number must be an integer.When ˘ κ is a non-integral rational number p/q ( p , q coprime) there is a close relationship betweenthe two quantities. A periodic arc will return toitself after being rotated q times by G p/q,T . Thisimplies the image of the curve is closed and thatthe Whitney turning number of its image is p .Conversely given a closed curve with Whitneyturning 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 isillustrated in figure 1. Example 1 - Epicyclic motion
Epicyclic motion was an ancient Greek modelfor our solar system . It was a fair approxima-tion for the motion of the planets as seen in theEarth’s rest frame but of course it has long sincebeen superseded by Heliocentric models.Epicyclic motion is formed by combining tworotary motions. A point on a circle, called the deferent , revolves with a constant angular veloc-ity ω . At each moment in time the revolvingpoint is the center for a another circle called the epicycle which spins about its center with con-stant angular velocity ω . A chosen point on theepicycle, that we will call the tracing point , stoodfor the location of a planet (see figure 2). Theorbit of a planet was represented by the curvegenerated by the tracing point.At any moment in time the location of thetracing point can be expressed as the sum of twovectors: a position vector for the location of thecenter of the epicycle relative to the center ofthe deferent and a position vector for the loca-tion of the tracing point relative to the center Prominent figures in the development of this modelwere Apollonius, Hipparchus, and Ptolemy. In Ptolemy’sversion the center was offset from the Earth but this didnot change the shape of the curve in the Earth’s restframe.
EpicycleDeferentTracing point
Figure 2: Epicyclic motion is the spin of theepicycle as its center revolves about the centerof the deferent. It can be expressed as the sumof two vectors which turn with fixed angular ve-locities.of the epicycle. We denote these two vectors attime t = 0 as ( x , y ) T and ( x , y ) T respectively.Figure 2 presents a simple time parametrizationfor curves traced out by epicyclic motion.Epicyclic motion can be generated by pro-jecting the orbits for a pair of uncoupled har-monic oscillators. Harmonic oscillators are con-servative systems. In this case the dynamics canbe obtained from the Hamiltonian H ( x , y , x , y ) = − (cid:0) ω (cid:0) x + y (cid:1) + ω (cid:0) x + y (cid:1)(cid:1) Because of the symmetry of this Hamiltonianit is arbitrary which pair of variables, x , x or y , y , are regarded as the positions and whichare regarded as the momenta. The choice onlyaffects the direction the system moves along theorbits. We choose x , x to be the position vari-ables and y , y to be conjugate momenta. Each( x j , y j ) T pair moves with angular frequency ω j along a circle in the state space.Projecting ( x , y , x , y ) T to the pair of posi-tion variables, ( x , x ) T , gives us Lissajous curves.We will consider Lissajous curves only briefly.This will occur in the following example on thespherical pendulum. To obtain epicyclic motion4rom uncouple harmonic oscillators we insteadproject ( x , y , x , y ) T to ( x + y , x + y ) T .We set r j = (cid:12)(cid:12)(cid:12)(cid:12) ( x j , y j ) T (cid:12)(cid:12)(cid:12)(cid:12) for j = 1 ,
2. For all t ∈ R we have | r − r | ≤ (cid:12)(cid:12)(cid:12)(cid:12) ( x ( t ) , y ( t )) T (cid:12)(cid:12)(cid:12)(cid:12) ≤ r + r We call | r − r | the minimum radius and r + r the maximum radius . The curve attains itsminimum radius when R ω t (( x , y ) T ) and R ω t (( x , y ) T ) point in opposite directions andit attains its maximum radius when they pointin the same direction. By suitably shifting timewe can suppose, without loss of generality, thatat t = 0 the vectors point in the same direction.This simplifies the time parameterization of thecurve to: (cid:18) x ( t ) y ( t ) (cid:19) = R ω t (cid:18) r (cid:19) + R ω t (cid:18) r (cid:19) (3.1)If ω = ω then the vectors R ω t ( r , T , R ω t ( r , T will continue to point in the samedirection and the tracing point will travel in acircle. Also if ω = 0 or ω = 0 the tracingpoint will travel in a circle so in this section weassume that ω ω ( ω − ω ) = 0. Furthermore,for the purpose of comparing these curves to thecurves generate by spiral tip meander in section4, we will assume ω > ω = ω the vectors R ω t ( r , T , R ω t ( r , T will alternately point in the sameand opposite directions. The condition that theypoint in the same or opposite direction is equiv-alent to || (cos( ω t ) , sin( ω t ) , × (cos( ω t ) , sin( ω t ) , || =sin(( ω − ω ) t ) = 0 Thus configurations for the deferent, epicycle,and tracing point which are congruent to the ini-tial condition occur with a periodicity of T =2 π/ | ω − ω | . It can be checked that this is thecommon minimal period for v ( t ) and κ ( t ). So wecan apply the theory from the previous sectionto epicyclic motion.The curves generated by the tracing point arenot properly called epicycles. These curves havenames based on a different construction method. They can be constructed as roulettes, i.e. byone curve rolling without slipping along anothercurve. One of the simplest non-trivial roulettesis generated by a point on a disk rolling withoutslipping along a line. These are called trochoids .The curves generated by epicyclic motion canbe generated by a circular disc rolling withoutslipping along another circular disc. This con-struction method was perhaps originally conceivedof by D¨urer in 1525 and then again by the as-tronomer Rømer in 1624. These curves havebeen studied by many mathematicians since.For some purposes it is useful to allow thetracing point to be outside of the rolling disc.We can treat the union of the rolling disc andthe tracing point as a single rigid body evenwhen their union does not form a connected set.This is done by applying the same motion ofthe rolling disc to the tracing point regardlessof where the tracing point happens to be. Andsince we are allowing the tracing point to be out-side of the rolling disc, we can dispense with thedisc’s interior in the definitions and just workwith a pair of circles, one fixed and one rollingalong other.The pair of circles are required to intersectin exactly one point. When they intersect in ex-actly one point they have the same tangent lineat the contact point, hence the circles are said tobe tangent to each other. If neither circle is in-side the other then they are said to be externallytangent. Otherwise they are said to be inter-nally tangent. If the fixed and rolling circles areexternally tangent then the curve generated bythe tracing point is called an epitrochoid . If thefixed circle is outside of the rolling circle thenthe curve is called a hypotrochoid . If the fixedcircle is inside of the rolling circle then the curveis called a peritrochoid .If the tracing point is inside of the rolling cir-cle then the hypotrochoid, epitrochoid, or per-itrochoid is said to be curtate . If the tracingpoint is outside of the rolling circle then the hy-potrochoid, epitrochoid, or peritrochoid is saidto be prolate . If the tracing point is on the Some authors reverse the meaning of curtate and pro-late, e.g. [4] hypocycloid , epicycloid ,or pericycloid respectivelyFigure 3: Three configurations for the fixed androlling circles. We assume the rolling circles re-volve anti-clockwise as indicated by the arrowsat their centers. This causes the rolling circleto turn clockwise for hypotrochoids and anti-clockwise for epitrochoids and peritrochoids asindicated by the arrows on the rolling circles.Its useful to have one term which encom-passes hypotrochoids, epitrochoids, and peritro-choids. Morely recognized this in 1894 and pro-posed using just “trochoids” [12] even thoughthis term is often restricted to the case where acircle rolls along a line. More recently the term“centered trochoid” has been proposed [3]. Weshall use central trochoid and use the term cen-tral cycloid to denote a hypocycloid, epicycloid,or pericycloid. The central cycloids are thosecentral trochoids that have cusps.Although the curves generated by epicyclicmotion are central trochoids the deferent and epicycle are generally not the same as the fixedand rolling circles in the roulette construction.However the fixed circle can be concentric withthe deferent and at any moment in time the rollingcircle can be concentric with the epicycle. Onlythe radii may need to be different.To determine the correct radii for the fixedand rolling circles we make use of the no slipcondition. The rolling circle may not slip as itgoes around the fixed circle. This means that theinstantaneous velocity of the contact point is thezero vector (0 , T . For epicyclic motion there isexactly one point, at any given moment in time,that is instantaneously at rest so this must bewhere the contact point for the fixed and rollingcircles are at that moment.The contact point of any pair of tangent cir-cles is collinear with their centers and by con-vention their centers are on the x -axis at t = 0so at this time the contact point is on the x -axisas well. We let a denote the x -coordinate of thecontact point at time t = 0. So | a | is the radiusof the fixed circle.At t = 0 the vector ( r , T is rotating withangular velocity ω about (0 , T while the vector( a, T − ( r , T is rotating with angular velocity ω about ( r , T . Taking the derivative withrespect to time, evaluating at t = 0, and settingthe result equal to the zero vector gives (cid:18) (cid:19) = ω R π/ (cid:18) r (cid:19) + ω R π/ (cid:18)(cid:18) a (cid:19) − (cid:18) r (cid:19)(cid:19) which has the unique solution a = (1 − ω /ω ) r (since ω = 0). To get the radius of the rollingwheel we set b = a − r , i.e. the directed distancefrom the center of the rolling circle to the contactpoint. The radius of the rolling circle is | b | .The center of the fixed circle is at (0 , T andat t = 0 the center of the rolling circle is at( r , T . If ω /ω < r < a , the contactpoint is on the right hand side of both circles,and the fixed circle is outside of the rolling cir-cle. So the curve is a hypotrochoid (see figure3). If 0 < ω /ω < < a < r , thecontact point is between the circles, and the cir-cles are externally tangent. So the curve is anepitrochoid. If 1 < ω /ω then a <
0, the con-tact point is on the left hand side of both circles,6nd the fixed circle is inside of the rolling circle.So the curve is a peritrochoid.From these facts the usual parameterizationsfor the central trochoids in terms of the radii | a | , | b | and the angle φ = ω t can easily be derived. Itis convenient to work with (3.1) because each ofthe epicyclic parameters occurs just once in theexpression. It is helpful to keep in mind, though,the different ways that central trochoids can beconstructed, e.g. either by epicyclic motion oras a roulette.Figure 4: The spaces of hypotrochoids, H , ofepitrochoids, E , and of peritrochoids, P . Alongwith the four distinguished cases, central cy-cloids, ellipses, billiard like, and rhodoneas.We are primarily concerned with the shape ofthe central trochoids rather than their position,orientation, or size. So we will determine thespace of similarity classes of central trochoids.For this purpose we introduce two geometricallyinvariant parameters for the central trochoids.One commonly used geometric invariant fordescribing the shape of a central trochoid is ρ = | ω /ω | r /r . This is often called the arm ratio .This is because the line segment connecting thecenter of the rolling circle to the tracing pointis often called the arm and ρ is the ratio of thearm’s length to the radius of the rolling circle, r / | b | = ρ . When ρ = 0 the resulting curve isjust a single point if ω = 0 and a circle other-wise. We do not wish to regard these as specialcases of central trochoids so we require ρ > < ρ < ρ = 1 the central trochoid is a central cy- cloid, and when ρ > χ = ω /ω (recall ω = 0). We call χ the turning ratio .It can also be expressed as χ = ( b/a ) / ( b/a − | b/a | is often called the wheel ratio .We will see that for curtate central trochoids thewheel ratio equals ˘ κ (for ω > ρ > χ = 0 , χ < < χ < < χ the central trochoid is a peritro-choid (see figure 4). We set H = { ( χ, ρ ) | χ < , ρ > }E = { ( χ, ρ ) | < χ < , ρ > }P = { ( χ, ρ ) | < χ, ρ > } and T = H ∪ E ∪ P .In addition to the central cycloids there arethree other cases of central trochoids worth dis-tinguishing. These cases also form curves in T as shown in figure 4. The first of these cases isgiven by the vertical line χ = −
1. The equationfor the hypotrochoid reduces to (cid:18) x ( t ) y ( t ) (cid:19) = (cid:18) ( r + r ) cos( ω t )( r − r ) sin( ω t ) (cid:19) which determines an ellipse with semi-major axis r + r and semi-minor axis | r − r | .We call the cases given by the diagonal lines ρ = | χ | “billiard like”. These curves have longarcs with low curvature alternating with shortarcs with high curvature. They are fairly wellapproximated by the paths made by a friction-less billiard ball rolling on a circular table whichtravels along straight line segments and bounceat the table’s edge.The remaining distinguished case of centraltrochoids corresponds to the central trochoid pass-ing through its own center of symmetry. Theminimum radius of a central trochoid is zero ifand only if the deferent and epicycle have thesame size. When r = r the distance of the7 κ hypo- epi- peri-prolate χ − − χ χ − curtate χχ − χ − χ χχ − Table 1: The value of ˘ κ for central trochoidswhen the rolling wheel revolves in the anticlock-wise direction, ω >
0. For ω < r cos((( ω − ω ) / t )which is essentially the defining condition for“rhodonea” curves . The condition r = r isequivalent to ρ | χ | = 1 which specifies a pair ofhyperbolic arcs in T (see figure 4). These hyper-bolic arcs form the subspace of rhodonea curves.Below the hyperbolic arcs the deferent is largerthan the epicycle while above the hyperbolic arcsthe deferent is smaller than the epicycle.Figure 5: The isogonal curves in T are verticalline segments. They are labeled with their valuefor ˘ κ . The value of ˘ κ changes by ± E isshaded to facilitate comparison to figure 4.In H the sets of hypocycloids, ellipses, bil-liard like curves, and rhodoneas all intersect atone point ( χ, ρ ) = ( − , . Forthe Tusi couple the tracing point goes back and Studied by Guido Grandi around 1723 Studied by Nasir al-Din al-Tusi around 1247 forth along a line segment. For the Tusi couplethe tangent vector is undefined at the end pointsof the line segment, as with the cusps of centralcycloids. The line segment can be regarded as adegenerate ellipse. The line segment is literallya billiard curve for a round table so we can sayit is a billiard like curve that passes through itscenter of symmetry.The Tusi couple is the only instance in whichthe curvature of a central trochoid is zero at anypoint. Hence central trochoids do not have in-flection points.The value of ˘ κ for central trochoids can becomputed from equation (2.2). It almost reducesto an uncomplicated function of χ except that itdepends on the signs of ω , ρ −
1, and χ ( χ − κ are shown in table 1. For anyreal number except zero there is a central tro-choid whose value for ˘ κ is the given real number.Combining χ = ( b/a ) / ( b/a −
1) with ˘ κ = χ/ ( χ −
1) gives ˘ κ = b/a . So we see from ta-ble 1 that ˘ κ = b/a for curtate hypotrochoidsand peritrochoids. For curtate epitrochoids ˘ κ = − b/a . The signs of a , b are the same for hy-potrochoids and peritrochoids and opposite forcurtate epitrochoids. Therefore the total turn-ing number for a periodic arc of a curtate cen-tral trochoid is the same as the wheel ratio, i.e. ˘ κ = | b/a | .The isogonal curves in T are vertical line seg-ments which span the heights of the curtate andprolate regions. When the line of central cycloidsis crossed the value of ˘ κ changes by ± χ and ρ . We willobtain the spaces of similarity classes of centraltrochoids by quotienting each of the spaces H , E , and P .In models for the planetary motion the epicy-cle was generally much smaller than the defer-ent. Actually though, because vector additionis commutative, it doesn’t matter which of thetwo circles we take to be the deferent and epicy-8igure 6: The space, H , of similarity classes ofHypotrochoids. The origin corresponds to theTusi couple. The four compass directions corre-spond to the four distinguished cases. The isogo-nal curves are shown in gray and they are labeledwith their | ˘ κ | values. The value of | ˘ κ | jumps as anisogonal curve passes through the line of hypocy-cloids.cle. If we let the center of the deferent revolveabout the center of the epicycle with angular ve-locity ω while the deferent spins with angularvelocity ω then the exact same curve can betraced out. However when we look for fixed androlling circles to generate the central trochoid weget a different location for the point at instanta-neous rest and thus different radii for the fixedand rolling circles. This fact is known as DavidBernoulli’s “dual generation theorem”.A consequence of the dual generation theo-rem is that the ordered pair (1 /χ, /ρ ) deter-mines the same similarity class of central tro-choids as ( χ, ρ ). A curtate hypotrochoid is geo-metrically similar to a prolate hypotrochoid and visa verse . A prolate epitrochoid is similar to acurtate peritrochoid and visa versa . A curtateepitrochoid is similar to a prolate peritrochoidand visa versa .The terms “hypotrochoid”, “epitrochoid”, “per-itrochoid”, “curtate”, and “prolate” do not des-ignate geometric similarity classes of curves but Figure 7: The space, E , of similarity classes ofEpitrochoids. The negative real axis and theorigin make up the complement of E in C . Theother three compass directions correspond to thethree distinguished cases of Epitrochoids. Theisogonal curves are shown in gray and they arelabeled with their | ˘ κ | values. The value of | ˘ κ | jumps as the isogonal curve passes through theline of epicycloids.merely describe how the curves can be constructedgeometrically. Many geometric figures can beconstructed in more than one way. This termi-nology can lead to some confusion particularlysince peritrochoids have been regarded as epitro-choids by some authors while other authors haveregarded them as hypotrochoids. Willson givesan overview on this terminology in the appendixto his 1898 book [18].Every central trochoid, except for central cy-cloids, can be constructed as a curtate centraltrochoid and as a prolate central trochoid. Thevalue of | ˘ κ | equals the wheel ratio in the cur-tate method of construction. We can think of˘ κ as a generalization of the wheel ratio to othertypes of curves with periodically varying curva-ture. Although ˘ κ can equal 0 for some curveswith periodically varying curvature but not forcentral trochoids.Suppose we have two central trochoids withparameters ω , ω , r , r and ω ′ , ω ′ , r ′ , r ′ . A9ecessary condition for two central trochoids tobe similar is for the ratio of their minimum radiusto their maximum radius be the same, i.e. | r − r | r + r = | r ′ − r ′ | r ′ + r ′ Since the radii are positive this equation is equiv-alent to the equation (cid:18) r ′ r ′ − r r (cid:19) (cid:18) r ′ r ′ − r r (cid:19) = 0If r ′ /r ′ = r /r then its necessary for ω ′ /ω ′ = ω /ω and if r ′ /r ′ = r /r then its necessary for ω ′ /ω ′ = ω /ω . Therefore the only points in T that correspond to the same similarity class as( χ, ρ ) is (1 /χ, /ρ ).We denote the space of similarity classes ofhypotrochoids by H . To denote the membersof H we use the term Hypotrochoid with the firstletter capitalized. The word “hypotrochoid” withall lower case letters describes how the curve wasconstructed. For ( χ, ρ ) ∈ H the function( χ, ρ ) (log( − χ ) + i log( ρ )) is onto the complex plane, C , and it is two toone everywhere except at the Tusi couple. Thepoint with the same image as ( χ, ρ ) is (1 /χ, /ρ )so we can identify H with C (see figure 6).The value of χ is negative for hypotrochoidsso regardless of whether it is prolate or curtate − < ˘ κ <
1. The projection from H to H mapsthe isogonal curves with ˘ κ = ± / H . We can associate the value of | ˘ κ | to each semiparabolic arc. The semiparabolicarcs whose values for | ˘ κ | sum to 1 form the en-tire parabola except for its vertex on the line ofHypocycloids.We denote the space of similarity classes ofepitrochoids by E . For members of E we usethe term Epitrochoid with the first letter capi-talized. The words “epitrochoid” and “peritro-choid” with all lower case letters describes howthe curve was constructed. For ( χ, ρ ) ∈ E ∪ P the function( χ, ρ ) (log( χ ) + i log( ρ )) is onto the complex plane C minus the non-positive real axis. It is two to one everywherein E ∪ P . The point with the same image as( χ, ρ ) is (1 /χ, /ρ ) so we can identify E with theslitted complex plane (see figure 7).The value of χ is positive for an epitrochoidor peritrochoid (for ω >
0) so regardless ofwhether it is prolate or curtate ˘ κ >
0. The pro-jection from E to E maps pairs of isogonal curveswith the same ˘ κ value to the same semiparabolicarc in E . To obviate the issue of how the Epitro-choids are parameterized we associate the valueof | ˘ κ | to each semiparabolic arc. If the differencein | ˘ κ | between a semiparabolic arc in the lowerhalf-plane of E and a semiparabolic arc in theupper half-plane of E is 1 then their union is theentire parabola except for its vertex on the lineof Epicycloids. Example 2 - The spherical pendulum
The spherical pendulum is an idealized mechan-ical system. It is comprised of a weightless in-extensible rod , essentially a line segment withlength ℓ . One end of the rod, the pivot , is mo-tionless for all time. The other end, the bob ,is the location of a point particle with mass m .The bob is constrained to move on a sphere ofradius ℓ centered at the pivot while subjected toa uniform gravitational field with strength g .We take the pivot to be the origin of a Carte-sian coordinate system for the spherical pendu-lum. We take the direction of the gravitationalfield to be the negative direction of the z -axis.The z -axis is also called the pendulum’s axis .The plane through the pivot and orthogonal tothe pendulum’s axis is the support plane . Thesupport plane contains the ( x, y )-axes of the co-ordinate system (see figure 8). Because the spher-ical pendulum is symmetrical about its axis theorientation of the ( x, y )-axes is completely arbi-trary.We also use spherical coordinates to specifythe position of the bob. Because the letters θ , ϕ are used through out this article to describe thedirection of the velocity along planar curves welet ϑ stand for the polar angle of the sphericalcoordinate system and ψ stand for the azimuthalangle (see figure 8).10 yz Figure 8: Cartesian coordinates, ( x, y, z ),and spherical coordinates, ( ϑ, ψ ) for the bobof a spherical pendulum. They are related by( x, y, z ) = ℓ (sin( ϑ ) cos( ψ ) , sin( ϑ ) sin( ψ ) , cos( ϑ )).The theory presented in section 2 can be gen-eralized to curves on a sphere using the conceptof geodesic curvature but to avoid unnecessarycomplications in this example we will study thepath of the bob from a bird’s eye view. More pre-cisely stated we orthogonally project the bob’spath on the sphere into the support plane. Wetreat the spherical pendulum as merely a mech-anism for generating planar curves which are theobjects of our study here.We present a short review of the sphericalpendulum’s dynamics. The bob moves in threedimensional space but it is subject to a singleholonomic constraint so it has two degrees offreedom. Two independent integrals of motionare the total energy or Hamiltonian, H and thevertical component of its angular momentum, J .The system is fully integrable. Its orbits are ei-ther quasiperiodic, periodic, or fixed points.The projection of the bob’s position to thependulum’s axis is always periodic. When thebob’s z -coordinate has a minimal period it issometimes referred to as the pendulum’s period even if the pendulum is behaving quasiperiodi-cally. So long as J = 0 the bob’s ( x, y ) coordi-nates will be rotated about the pendulum’s axisby a nonzero amount during the pendulum’s pe-riod. Typically the angle for this rotation is ir-rational so the overall motion and its projection to the support plane is quasiperiodic. It will beshown that the pendulum’s period is the com-mon minimal period of v ( t ), κ ( t ) for the ( x, y )-curves so we can apply the theory from section2. The kinetic energy of the bob is12 m ( ˙ x + ˙ y + ˙ z ) = 12 mℓ ( ˙ ϑ + sin ( ϑ ) ˙ ψ )and the potential energy is U = mgz = mgℓ cos( ϑ ).Since U is independent of ˙ ϑ and ˙ ψ the conjugatemomenta are P ϑ = ∂∂ ˙ ϑ mℓ ( ˙ ϑ + sin ( ϑ ) ˙ ψ ) = mℓ ˙ ϑJ = ∂∂ ˙ ψ mℓ ( ˙ ϑ + sin ( ϑ ) ˙ ψ ) = mℓ sin ( ϑ ) ˙ ψ These are the horizontal and vertical compo-nents of the bob’s total angular momentum re-spectively. The state of the spherical pendulumis completely specified by the canonical variables( ϑ, P ϑ , ψ, J ). The Hamiltonian is H ( ϑ, P ϑ , ψ, J ) = 12 mℓ (cid:18) P ϑ + J sin ( ϑ ) (cid:19) + mgℓ cos( ϑ )(3.2) The fact that H is independent of ψ showsus that the value of J is constant. Althoughthe state space is four dimensional the dynam-ics can be reduced to two dimensions because H , J are constant. Moreover the reduced sys-tem has a nondimensionalized form. Physically,the parameters, m , g , ℓ , are limited to positivevalues and varying them does not produce anyqualitative changes in behavior so long as theyremain positive. To obtain the reduced systemwe nondimensionalized the constants of motion,we define a dimensionless potential energy andits dimensionless rate of change, and we define adimensionless time:( h, j ) = (cid:18) Hmgℓ , Jmℓ √ gℓ (cid:19) ( u, w ) T = Umgℓ , s ℓg ˙ u ! T t = p g/ℓ t The equations of motion which can be obtainedfrom the Hamiltonian (3.2) can be used to show11hat ( u, w ) T satisfies the differential equation (cid:18) du/d t dw/d t (cid:19) = (cid:18) w u − hu − (cid:19) (3.3)This is known as the reduced system for the spher-ical pendulum. Its is not a straight forward ini-tial value problem. First of all the initial valuefor u must be in the interval [ − , h = 12 w + j − u + u (3.4)Although ( u, w ) T varies with time the value of h depends on ( u, w ) T in such a way that it does notchange with time. The value of h can be deter-mined 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 conditionsit can be treated as a fixed parameter in (3.3).The ( x, y )-curves can be obtained from a so-lution for u by using a single quadrature. ψ ( t ) = ψ (0) + j Z √ g/ℓ t dτ − u ( τ ) (cid:18) x ( t ) y ( t ) (cid:19) = ℓ p − u ( t ) (cid:18) cos( ψ ( t ))sin( ψ ( t )) (cid:19) (3.5)The maximum potential energy is attainedwhen the bob is directly above the pivot and theminimum potential energy is attained when thebob is directly below the pivot. If the bob has nokinetic energy when its directly above or belowthe pivot then it will remain where it is. Thesetwo states are the fixed points of the Hamiltoniansystem (3.2). If h = − h . There is no limitto how fast the bob can move so h has no upperbound.For each h ≥ − j . The extreme values for j can beobtained by rearranging (3.4) to − j = 12 w − ( h − u )(1 − u ) (3.6)and differentiating the right hand side with re-spect to u and w while treating h as a fixed pa-rameter. There is one critical point for j which j Figure 9: The energy-momentum space, P , forthe spherical pendulum is the light gray regionbounded by the black ± j max curves. The isogo-nal curves, shown in dark gray, are labeled withtheir values for ˘ κ . They radiate from (1 , j = 2 h (9 − h ), is where in-flection points partition the corresponding ( x, y )-curves into periodic arcs. Outside of the oval the( x, y )-curves do not have inflection points. Insetsshow ( x, y )-curves in gray each with a periodicarc shown in black. Dotted arrows point to thecorresponding ( h, j ) values.is (( h − √ h + 3) / , T . The maximum valuefor j is j max = 29 q h + 3) / − h + 9 h )and the minimum value is − j max . The graph of j max is an increasing, concave down, curve witha single end point as shown in figure 9. Theset of all possible values for ( h, j ) is the energy-momentum space for the spherical pendulum, P = { ( h, j ) : h ≥ − , | j | ≤ j max } For j = 0 the angular velocity ˙ ψ is alwayszero and the pendulum moves within a vertical12lane. In this case it is often called a planar pen-dulum even though there are no physical forcesconstraining it within a plane. If ( h, j ) = ( − , x, y )-curve is just a point. If ( h, j ) = (1 , h such that h > − h = 1 all of the( x, y )-curves corresponding to ( h,
0) are line seg-ments with the same length.The right hand side of (3.6) can be taken asa Hamiltonian function for the reduced system(3.3). Thus the critical point of j correspondsto a fixed point of the reduced system. So theextreme values for j are attained when w = 0, i.e. the potential energy is constant. In thesecases the height of the bob does not change. Thisis often referred to as a conical pendulum becausethe rod sweeps out a cone. The bob rotates alonga horizontal circle with the sign of j determiningthe direction of rotation. For each h > − x, y )-curves determined by ( h, ± j max ) arethe same circle.For each ( h, j ) in R = { ( h, j ) : h > − , < | j | < j max } the subset of the spherical pendulum’s state spacewhich is mapped to ( h, j ) is a torus [2]. Theaction of the group of rotations about the pen-dulum’s axis can be extended to the whole statespace of the pendulum and for each ( h, j ) ∈ R the corresponding torus is invariant under thisaction. Rotations about the pendulum’s axishave no effect on w and these tori can be pro-jected into a three dimensional space while pre-serving their symmetry by using the coordinates( x, y, w ) (see figure 10). For each ( h, j ) ∈ R all of the corresponding ( x, y )-curves are rotatedcopies of each other. The orbits typically windquasiperiodically around the torus but they canbe periodic. In either case the curvature of the( x, y )-curve varies periodically. The value of ˘ κ isdefined for the corresponding ( x, y )-curve if andonly if ( h, j ) ∈ R .Since the shape of the ( x, y )-curves is com-pletely determined by ( h, j ) ∈ R we can think of˘ κ as a function on the space R . We let T denote Cushman’s R space is a little larger than this. Figure 10: The projection of the invariant toricorresponding to given ( h, j ) values into the co-ordinate space for ( x, y, w ). The location of the( h, j ) values in P is shown in figure 9. Eachtorus shows the projection of a rotationally sym-metric quasiperiodic orbit in the torus.the period of the reduced system in dimension-less time t . T depends on ( h, j ). In physical time t the period is p ℓ/g T . We obtain the velocityof the ( x, y )-curves from (3.5) and (3.6). v = p gℓ (2 h − u − w )The period of 2 h − u − w is the same as theperiod of the reduced system and since the ex-pression under the radical is positive for all timefor ( h, j ) ∈ R the period of v is the same as theperiod of the reduced system. We also obtainthe curvature from (3.5) and (3.6). κ = ˙ x ¨ y − ˙ y ¨ xv = j p g ℓ h − uv which has the same period. From equation (2.2)and the change of variables theorem we get anexpression for ˘ κ entirely in terms of the reduced13ystem. ˘ κ = j π Z T h − u h − u − w dτ (3.7)A few numerically computed isogonal curves in R are shown in figure 9. The isogonal curvesradiate from the point ( h, j ) = (1 , h, j ) is near thevertex ( − , ∈ P by linearizing the systemabout its fixed point for ( h, j ) = ( − , x, y )-curves forthe spherical pendulum could be well approxi-mated by Lissajous curves since we are project-ing the state variables to the position variables( x, y ). However because the spherical pendulumis symmetrical about its axis the two frequenciesof the linearized system are equal and therefore,regardless of the initial conditions, the only typeof Lissajous curves generated by projecting thestate variables of the linearized system to the po-sition variables are ellipses or line segments. Fur-thermore the eigenvalues of the linearized systemare purely imaginary so the Hartman-Grobmantheorem does not apply [5], i.e. there need not bea neighborhood in which the spherical pendulumis equivalent to its linearization. The sphericalpendulum is in fact highly nonlinear [11]. Nomatter how close ( h, j ) ∈ R is to ( − ,
0) the( x, y )-curves resemble Hypotrochoids more thanthey do Lissajous curves (see insets in figure 9),which is not too surprising given the geometryof the spherical pendulum.It is interesting to compare the spaces P and R (figure 9) with the spaces H (figure 5) and H (figure 6). For these spaces the reflection aboutthe horizontal axis maps isogonal curves to isog-onal curves and the range of observed values for˘ κ is in the open interval ( − , | ˘ κ | belowthe horizontal axis is the open interval (1 / , R thevalue of | ˘ κ | can be within any tiny distance above1 / / h ∈ ( − , κ appears to converge to 1 / j ap-proaches 0 from above while it appears to con-verge to − / j approaches 0 from below. Nu-merical analysis also indicates that the value of | ˘ κ | can be within any tiny distance below 1 butit can not be 1 or more. For h > κ appears to converge to 1 as j approaches 0from above while it appears to converge to − j approaches 0 from below. As ( h, j ) crosses thehorizontal axis of P the ( x, y )-curves transitionby collapsing to a line segment.In H the value of ˘ κ converges to 1 / χ, ρ )approaches the Tusi couple, ( − , − / χ, ρ ) approachesthe Tusi couple from above. As ( χ, ρ ) crossesthe Tusi couple the ( x, y )-curves transition bycollapsing to a line segment.In P there is a half-line for which the cor-responding ( x, y )-curves are line segments whilein H there is a single point for which the ( x, y )-curves is a line segment. On the other hand in P the isogonal curves radiant from a single pointwhile in H all of the isogonal curves pass througha half-line.There are some important differences betweenin P and H . The value of ˘ κ for billiard like Hy-potrochoids must be in the interval [ − / , / x, y )-curves generated by the sphericalpendulum do not resemble billiard like Hypotro-choids. They also do not resemble rhodonea curvessince they only pass through their center of sym-metry when they collapse to line segments. Andunlike Hypotrochoids an ( x, y )-curve for the spher-ical pendulum can have inflection points. Thishappens when ( h, j ) is inside the oval shown infigure 9.The value of ˘ κ is associated with the intrin-sic precession of the spherical pendulum. Themotion of the projected image of the bob in thesupport plane can be thought of as a compoundmotion of a point around an ellipse with the ro-tation of the ellipse about its center. The intrin-sic precession of the spherical pendulum is therotation of the ellipse.It should be briefly pointed out that the in-trinsic precession of a spherical pendulum is dis-tinct from Foucault precession. The differencewas recognized by Foucault himself. As is wellknown, Foucault designed and built a sphericalpendulum in 1851 to measure the rotation of theEarth [9]. Foucault’s pendulum was designed tobe set librating within a vertical plane. Since the14endulum’s support is rotating with the Earththe plane of libration appears to rotate relativeto the ground below it. This motion is called Foucault precession .Foucault found that it can be difficult to starta spherical pendulum with sufficiently little an-gular momentum so that it will appear to oscil-late within a vertical plane. Even a small amountof angular momentum led to an intrinsic preces-sion comparable to the Foucault precession. Toovercome this apparent “instability” he designedhis pendulum with very large m and ℓ .If | j | is small enough the periodic arcs of the( x, y )-curve generated by the spherical pendulumcan be fairly well approximated by the periodicarcs of an ellipse (see top left inset in figure 9).The approximating ellipse turns in the same di-rection as the bob so the total curvature of aperiodic arc of the ( x, y )-curve of the sphericalpendulum is the sum of the total curvature ofa periodic arc of the approximating ellipse withthe intrinsic precession of the approximating el-lipse. For any nondegenerate ellipse the value ofthe total curvature of a periodic arc is ˘ κ = ± / (cid:26) ˘ κ − / κ > / κ + 1 / κ < − / | j | is large the periodic arcs of the ( x, y )-curveare not well approximated by periodic arcs ofan ellipse (see bottom left inset in figure 9). Inthese cases it is not very helpful to think of the( x, y )-curve as being generated by a precessingellipse.The value of ˘ κ is defined for all ( h, j ) ∈ R regardless of how poorly the periodic arcs of an( x, y )-curve can be approximated by the periodicarcs of an ellipse. The value of ˘ κ tells us howfar the ( x, y )-curve turns during the pendulum’speriod as well as providing us with informationabout the symmetry of the ( x, y )-curve. In this section we consider two related models fornatural systems that generate curves with peri- odic curvature. These are the Shenoy-Rutenbergmodel for the paths taken by the bacterium
Lis-teria monocytogenes in eukaryotic cells [6] andthe Barkley-Kevrekidis model for the meanderof spiral waves in excitable media such as theBZ reaction [1].
Example 3 - Actin based motility
L. monocytogenes transport themselves in eu-karyotic cells by catalyzing the polymerizationof the cytoskeletal protein actin. This methodof propulsion can result in a bacterium followinga complicated path within the cytosol at a fairlyconstant speed. The curvature of the paths tendsto vary periodically with time so the curvatureand speed have a common minimal period andwe can apply the theory from section 2.Recall from section 2 that the velocity’s ori-entation is ϕ ( t ) = κ t + e ϕ ( t ) where e ϕ ( t ) is pe-riodic. In the Shenoy-Rutenberg model e ϕ ( t ) =(Ω /ω ) sin( ω t ) where ω is the angular frequencyof the bacterium’s spin about its long axis and Ωis a monotonic function of the distance of the ef-fective propulsive force from the long axis. In themodel the speed of the bacterium is the constant v . The common minimal period of the curva-ture and speed is T = 2 π/ω and so κ = ω ˘ κ .The points of maximal curvature occur for t ∈ (2 π/ω ) Z and the points of minimal curvatureoccur for t ∈ ( π/ω ) + (2 π/ω ) Z .Since (Ω /ω ) sin( ω t ) is an odd function thefull image of the curve has reflectional symme-try. Joining an arc over a half-period with itsreflected image gives a periodic arc. For integral˘ κ the rest of the curve can be obtained by trans-lating the periodic arc. For non-integral ˘ κ therest of the curve can be obtained by rotating theperiodic arc about ( x, y ) T .When the initial direction is horizontal, i.e. ϕ (0) = 0, and the starting point is at the origin, i.e. ( x (0) , y (0)) T = (0 , T , the time parameter-ization for the curve is (cid:18) x ( t ) y ( t ) (cid:19) = v Z t (cid:18) cos( ω ˘ κτ + (Ω /ω ) sin( ω τ ))sin( ω ˘ κτ + (Ω /ω ) sin( ω τ )) (cid:19) dτ (4.1) There is no closed form for this integral in termsof elementary functions but we can obtain a closedform that accurately approximates it by borrow-ing a technique from civil engineering known as15igure 11: Generating a spiral easement approx-imation for an ( x, y )-curve. Above: A clothoid.It is symmetrical under a half turn and the cen-ter is its unique inflection point. Below: Theapproximating ( X , Y )-curve. The two arcs high-lighted in black are geometrically similar to eachother. The ( X , Y )-curve can be obtained by suc-cessively reflecting its black arc about the fivemirror lines shown in figure 12.spiral easement. This technique varies the cur-vature of roads and train tracks in a piecewiselinear fashion. For the Shenoy-Rutenberg modelwe approximate the curvature with the triangu-lar wave form κ ( t ) ≈ ω ˘ κv + Ω v (cid:18) π arcsin(cos( ω t )) (cid:19) This gives a piecewise quadratic approximationfor e ϕ ( t ): Ω ω (cid:18) π arcsin(cos( ω t/ ω t/ (cid:19) The rotational symmetry of the curve is unaf- fected by this approximation since only the e ϕ ( t )term in ϕ ( t ) is altered and the reflectional sym-metry is unaffected since e ϕ ( t ) remains an evenfunction. Because of the symmetry the errorvaries periodically and so remains bounded. Thequantity (cid:12)(cid:12)(cid:12)(cid:12) π arcsin(cos( ω t/ ω t/ − sin( ω t ) (cid:12)(cid:12)(cid:12)(cid:12) is never more than 3 . o at any time. Figure 12shows an example of a spiral easement approx-imation for the parametrized curve in equation(4.1)We let ( X ( t ) , Y ( t )) T denote the time param-eterization for the approximating curve to (4.1).On the interval [0 , π/ω ] the curvature can bewritten as the linear polynomial 2 β ( α − βt ) where α = πω ˘ κ + 4Ω2 πβv β = 2 π r ω Ω v Integrating over a sub-interval [0 , t ] ⊆ [0 , π/ω ]gives us a closed form time parameterization foran arc of the ( X ( t ) , Y ( t ))-curve, (cid:18) X ( t ) Y ( t ) (cid:19) = v β R ( α ) (cid:18) − C ( α − βt ) + C ( α ) S ( α − βt ) − S ( α ) (cid:19) (4.2) where C ( t ), S ( t ) are the Fresnel trigonometricfunctions.Figure 12: An ( x, y )-curve with (˘ κ, Ω /ω ) =(4 / ,
1) along with its approximating ( X , Y )-curve translated to have the same center. Thecurves’ five mirror lines are also displayed.16he planar curve t ( C ( t ) , S ( t )) T is knownas a clothoid . It is shown at the top of figure11. It has unit speed so t is the arc length fromit center, (0 , T to ( C ( t ) , S ( t )) T . The curva-ture at ( C ( t ) , S ( t )) T is 2 t so every possible curva-ture occurs at exactly one point of the clothoid.Since the curvature is monotonic the clothoiddoes not intersect itself and since the curvatureis unbounded in the positive and negative direc-tions the clothoid spirals around two points.The right hand side of equation (4.2) is theapplication of a odd similarity transformation toan arc of the clothoid. This approximation tech-nique amounts to taking the extremal curvaturesof the ( x, y ) T -curve, determining the two pointson the clothoid where these extremal curvaturesoccur, and applying an odd similarity to the arcin the clothoid connecting the points of extremalcurvature as shown in figure 11. The rest of the( X , Y )-curve is obtained by the action of thesymmetry group of the ( X , Y )-curve. The ap-proximation can be further refined by translatingthe ( X , Y )-curve so that it has the same centeras the ( x, y )-curve (see figure 12).The value of ˘ κ determines the rotational sym-metry of the curve. The value of Ω /ω deter-mines the length of the clothoid arc used in ap-proximating the curve. The effect of Ω /ω forfixed ˘ κ is perhaps best illustrated in the ˘ κ = 1 / /ω varies the points of minimal curva-ture oscillate in unison along mirror lines andthe points of maximal curvature oscillate in uni-son along mirror lines. For small Ω /ω the curvehas an oval shape with the points of minimalcurvature closer to the center than the pointsof maximal curvature. As Ω /ω increases thepoints of minimal curvature move toward thecenter and inflection points appear. Next thepoints of minimal curvature pass through thecenter together and then move outward. Eventu-ally the points of minimal curvature go far fromthe center while the points of maximal curva-ture go near the center. The points of maximal It is also known as Euler’s spiral and as Cornu’sspiral. curvature pass through the center together andthen proceed outward. Afterwards the points ofminimal curvature pass back through the centeragain. The process is reminiscent of a loom ex-cept the curve becomes wound up around fourpoints (when ˘ κ = 1 /
2) instead of being woven.Figure 13: ( X , Y )-curves with fixed ˘ κ = 1 / /ω . Each row shows a pair of pointswith extremal curvature passing through eachother at the center thereby introducing a pair ofcrossing points which persist as Ω /ω continuesto increase.More generally, for ˘ κ / ∈ Z , the points of max-imal curvature coincide with the center if andonly if ( X (0) , Y (0)) T = ( X , Y ) T while the pointsof minimal curvature coincide with the center ifand only if ( X ( π/ω ) , Y ( π/ω )) T = ( X , Y ) T . Ifthe points of minimal curvature coincide with thecenter then cos( α ) ( C ( γ ) − C ( α )) + sin( α ) ( S ( γ ) − S ( α )) = 0(4.3) where γ = α − βπ/ω . This gives us a conditionwhich must hold between ˘ κ and Ω /ω . The samecondition holds if the points of maximal curva-ture coincide with the center except ˘ κ is replacedwith − ˘ κ . These two conditions determine twosets of curves in the (˘ κ, Ω /ω ) parameter planewhich are shown in figure 14.These two sets of curves only intersect at in-teger values of ˘ κ . It turns out in these cases that17he congruence G ˘ κ,T reduces to the identity map,that the image of the ( X , Y )-curve is closed, andthat ˘ κ is its turning number.Figure 14: The parameter plane for equation(4.2) and the curves defined by (4.3). Thesolid black curves correspond to ( X , Y )-curveswhose points of minimal curvature coincide withthe center. The dotted black curves correspond( X , Y )-curves whose points of maximal curva-ture coincide with the center. The solid blackand dotted black curves only intersect on the ver-tical gray lines which correspond to integer val-ues for ˘ κ . The gray diagonal lines correspond tothe appearance of inflection points in the ( X , Y )-curves. Example 4 - Spiral tip meander
Barkley’s model [1] for the spiral wave tip me-ander can be written as the ordinary differentialequation ˙ x ˙ y ˙ ϕ ˙ v ˙ w = v cos( ϕ ) v sin( ϕ ) γ wv ( − / / v + α w − v ) w ( − v − w ) Here we have replaced the variable name ‘ s ’ inBarkley’s equations (3) and (4) with the variablename ‘ v ’ in keeping with the convention in thisarticle that s stands for arc length and v standsfor speed. From equation (2.2) the total curvature perperiodic arc is˘ κ ≈ √ νπ Z √ π/ w ( τ ) dτ where ν = γ / √
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