Moving frame and integrable system of the discrete centroaffine curves in R 3

Transcription

1 Movig frame ad itegrable system of the discrete cetroaffie curves i R 3 Yu Yag, Yahua Yu Departmet of Mathematics, Northeaster Uiversity, Sheyag 0004, P R Chia arxiv: v2 [mathdg] 27 Nov 206 Abstract Ay two equivalet discrete curves must have the same ivariats at the correspodig poits uder a affie trasformatio I this paper, we costruct the movig frame ad ivariats for the discrete cetroaffie curves, which could be used to discrimiate the same discrete curves from differet graphics, ad estimate whether a polygo flow is stable or periodically stable I fact, usig the similar method as the Freet-Serret frame, a discrete curve ca be uiquely idetified by its cetroaffie curvatures ad torsios I 878, Darboux studied the problem of midpoit iteratio of polygos[2] Berlekamp et al studied this problem i detail[2] Now, through the cetroaffie curvatures ad torsios, the iteratio process ca be clearly quatified Exactly, we describe the whole iteratio process by usig cetroaffie curvatures ad torsios, ad its periodicity could be directly exhibited As a applicatio, we would obtai some stable discrete space curves with chageless curvatures ad torsios after multistep iteratio For the petagram map of a polygo, the affiely regular polygos are stable Furthermore, we fid the covex hexagos with parallel ad equi-legth opposite sides are periodically stable, ad some covex parallel ad equi-legth opposite sides octagos are also periodically stable The proofs of these results are obtaied usig the structure equatios of the discrete cetroaffie curves ad the itegrable coditios of its flows MSC 200: 52C07, 53A5 Key Words: Discrete differetial geometry, affie trasformatio, discrete curvature flow Itroductio Discrete differetial geometry has attracted much attetio recetly, maily due to the growth of computer graphics Oe of the mai issues i discrete differetial geometry is to defie suitable discrete aalogous of the cocepts of smooth differetial geometry[4, 6] More recetly, the expasio of computer graphics ad applicatios i mathematical physics have give a great impulse to the issue of givig discrete equivalets of affie differetial geometric objects[3, 9, 0] I [5] a cosistet defiitio of discrete affie spheres is proposed, both for defiite ad idefiite metrics ad i [2] a similar costructio is doe i the cotext of improper affie spheres This work was supported by NSFC(Nos ad 37080) Correspodig author addresses: yagyu@maileueduc (Y Yag), yuyahua@maileueduc(y Yu)

2 Group based movig frames have a wide rage of applicatios, from the classical equivalece problems i differetial geometry to more moder applicatios such as computer visio[20, 25] The first results for the computatio of discrete ivariats usig group based movig frames were give by Olver[22] who calls them joit ivariats; moder applicatios to date iclude computer visio[23] ad umerical schemes for systems with a Lie symmetry[8] Movig frames for discrete applicatios as formulated by Olver do give geeratig sets of discrete ivariats, ad the recursio formulas for differetial ivariats are so successful for the applicatio of movig frames to calculus-based applicatios Recet developmet of a theory of discrete equivariat movig frames has bee applied to itegrable differetial-differece systems[20, 25] Followig the ideas of Klei, preseted i his famous lecture at Erlage, several geometers i the early 20th cetury proposed the study of curves ad surfaces with respect to differet trasformatio groups I geometry, a affie trasformatio, affie map or a affiity is a fuctio betwee affie spaces which preserves poits, straight lies ad plaes Also, sets of parallel lies remai parallel after a affie trasformatio A affie trasformatio does ot ecessarily preserve agles betwee lies or distaces betwee poits, though it does preserve ratios of distaces betwee poits lyig o a straight lie Examples of affie trasformatios iclude traslatio, scalig, homothety, similarity trasformatio, reflectio, rotatio, shear mappig, ad compositios of them i ay combiatio ad sequece A cetroaffie trasformatio is othig but a geeral liear trasformatio R x Ax R, where A GL(,R) I 907 Tzitzéica foud that for a surface i Euclidea 3-space the property that the ratio of the Gauss curvature to the fourth power of the distace of the taget plae from the origi is costat is ivariat uder a cetroaffie trasformatio The surfaces with this property tur out to be what are ow called Tzitzéica surfaces, or proper affie spheres with ceter at the origi I cetroaffie differetial geometry, the theory of hypersurfaces has a log history The otio of cetroaffie miimal hypersurfaces was itroduced by Wag [3] as extremals for the area itegral of the cetroaffie metric See also [33, 34] for the classificatio results about cetroaffie traslatiosurfaces adcetroaffie ruled surfaces ir 3 Several authors studied the curves uder cetroaffie trasformatio group usig some methods([3, 5], etc), ad Liu defie cetroaffie ivariat arc legth ad cetroaffie curvature fuctios of a curve i affie -space directly by the parameter trasformatios ad the cetroaffie trasformatios[8] Usig the equivariat method of movig frames, Olver costructed the explicit formulas for the geeratig differetial ivariats ad ivariat differetial operators for curves i 2-dimesioal ad 3-dimesioal cetro-equi-affie ad cetroaffie geometry ad surfaces i 3-dimesioal cetro-equi-affie geometry[24] The study of discrete itegrable systems is rather ew It bega with discretizig cotiuous itegrable systems i 970s The best kow discretizatio of the Korteweg-de Vries equatio (KdV) is the Toda lattice[30] Aother famous itegrable discretizatio of the KdV equatio is the Volterra lattice[6, 9] I this paper, the defiitios ad costructios of discrete itegrable systems are ature ad useful It arises as aalogues of curvature flows for polygo evolutios I mathematics, curvature refers to ay of a umber of loosely related cocepts i differet areas of geometry Ituitively, curvature is the amout by which a geometric object deviates from beig flat, but this is defied i differet ways depedig o the cotext The arragemet of the paper is as follows: I Sect 2 we recall the basic theory ad otios for cetroaffie differetial geometry, the basic otatios for discrete curves ad cetroaffie curves There are some results with cetroaffie curvatures ad torsios for cetroaffie plaar curves ad space curves i Sect 3 ad Sect 4 I Sect 5, we exted the curve shorteig flow to the discrete cetroaffie curve I Sects 6 ad 7, we study the trasversal flow ad taget flow for a discrete cetroaffie curve respectively Some iterest examples for polygo iteratio 2

3 are show Fially, Sect 8 describes some applicatios of cetroaffie curvatures ad torsios We coclude with idicatios of future work 2 Affie mappigs ad trasformatio groups, basic otatios If X ad Y are affie spaces, the every affie trasformatio f : X Y is of the form x M x+ b, where M is a liear trasformatio o X ad b is a vector i Y Ulike a purely liear trasformatio, a affie map eed ot preserve the zero poit i a liear space Thus, every liear trasformatio is affie, but ot every affie trasformatio is liear For may purposes a affie space ca be thought of as Euclidea space, though the cocept of affie space is far more geeral (ie, all Euclidea spaces are affie, but there are affie spaces that are o-euclidea) I affie coordiates, which iclude Cartesia coordiates i Euclidea spaces, each output coordiate of a affie map is a liear fuctio (i the sese of calculus) of all iput coordiates Aother way to deal with affie trasformatios systematically is to select a poit as the origi; the, ay affie trasformatio is equivalet to a liear trasformatio (of positio vectors) followed by a traslatio It is well kow that the set of all automorphisms of a vector space V of dimesio m forms a group We use the followig stadard otatios for this group ad its subgroups([7]): GL(m,R) := {L : V V L isomorphism}; SL(m,R) := {L GL(m,R) detl = } Correspodigly, for a affie space A,dimA = m, we have the followig affie trasformatio groups A(m) := {α : A A L α regular} is the regular affie group S(m) := {α A detα = } is the uimodular(equiaffie) group Z p (m) := {α A α(p) = p} is the cetroaffie group with ceter p A τ(m) := {α : A A there exists b(α) V,st pα(p) = b(α), p A} is the group of trasformatios o A Let G be oe of the groups above ad S,S 2 A subsets The S ad S 2 are called equivalet modulo G if there exists a α G such that S 2 = αs The stadard properties of affie mappigs come from the properties of the associated liear mappig Recall i particular: (i) Parallelism is ivariat uder affie mappigs (ii) The partitio ratio of 3 poits is ivariat uder affie mappigs (iii) The ratio of the volumes of two parallelepipeds is affiely ivariat Moreover, covexity is a affie property 3

4 Theorem 2 α : A A is a regular affie trasformatio if ad oly if α is bijective, cotiuous ad preserves covexity (For a more geeral result see [32] ) I cetroaffie geometry we fix a poit i A(the origi O A without loss of geerality) ad cosider the geometric properties i variat uder the cetroaffie group Z p Thus the mappig π 0 : A V idetifies A with the vector space V ad Z O with GL(m,R) We will start very simple, by discretizig the otio of a smooth curve That is, we wat to defie a discrete aalog to a smooth map from a iterval I R to R By discrete we mea here that the map should ot be defied o a iterval i R but o a discrete (ordered) set of poits therei It turs out that this is basically all we eed to demad i this case: Defiitio 22 Let I Z be a iterval (the itersectio of a iterval i R with Z, possibly ifiite) A map r : I R is called a discrete curve, whe we put the startig poit of the vector r to the origi O R Obviously, a discrete curve is a polygo A discrete curve r is said to be periodic (or closed) if I = Z ad if there is a p Z such that r(k) = r(k +p) for all k I The smallest possible value of p is called the period I fact, we ca defie (2) r(t) = (t k) r(k)+(k + t) r(k +), t (k,k +),k Z The simplicity of a smooth curve ca be geeralized to the discrete case Defiitio 23 A closed curve r is simple if it has o further self-itersectios i oe period; that is, if t,t 2 [k,k + p),t t 2,k Z, the r(t ) r(t 2 ) We call a o closed curve is simple if t,t 2 [a,b),t t 2, the r(t ) r(t 2 ) For coveiece, sometimes we will write r k = r(k) ad eve r = r k, r = r k+, ad r = r k Defiitio 24 The edge taget vector of a discrete curve r : I R is defied as the forward differece t k := r k+ r k We could have writte t := r r as well The lies passig through the termial poits of r k ad r k+, k I Z are called taget lies of the discrete curve r With these preparatios, it is ature to give the defiitio of covexity for a discrete plaar curve as follows Defiitio 25 A covex discrete curve is the curve i the plae which lies completely o oe side of each ad every oe of its taget lies I classical differetial geometry of curve, the followig results are well-kow, ad we will have some similar results for a discrete curve with cetroaffie curvatures ad torsios i the ext two sectios Theorem 26 A closed regular plaar simple curve C is covex if ad oly if its curvature is either always o-egative or always o-positive, ie, if ad oly if the turig agle (the agle of the taget to the curve) is a weakly mootoe fuctio of the parametrizatio of the curve(for details see [4]) 4

5 I classical differetial geometry, the Freet-Serret formulas describe the kiematic properties of a particle movig alog a cotiuous, differetiable curve i three-dimesioal Euclidea space R 3, or the geometric properties of the curve itself irrespective of ay motio More specifically, the formulas describe the derivatives of the so-called taget, ormal, ad biormal uit vectors i terms of each other The taget, ormal, ad biormal uit vectors, ofte called T, N, ad B, or collectively the Freet-Serret frame, together form a orthoormal basis spaig R 3 ad are defied as follows: T is the uit vector taget to the curve, poitig i the directio of motio N is the ormal uit vector, the derivative of T with respect to the arclegth parameter of the curve, divided by its legth B is the biormal uit vector, the cross product of T ad N The Freet-Serret formulas are: ( dt ds, dn ds, db ds ) = (T,N,B) 0 κ 0 κ 0 τ 0 τ 0 where d is the derivative with respect to arc legth, κ is the curvature, ad τ is the torsio of ds the curve The two scalars κ ad τ effectively defie the curvature ad torsio of a space curve The associated collectio, T, N, B, κ ad τ, is called the Freet-Serret apparatus Ituitively, curvature measures the failure of a curve to be a straight lie, while torsio measures the failure of a curve to be plaar At the same time we have Theorem 27 (Fudametal theorem of space curves) Let r (s) ad r 2 (s) be two vectorvalued fuctios that represet the space curves C ad C 2 respectively, ad suppose that these curves have the same o-vaishig curvature κ(s) ad the same torsio τ(s) The C ad C 2 are cogruet such that each ca be rigidly shifted/rotated so that every poit o C coicides with every poit o C 2 (for details see [4]) Before startig ext sectio, we shall eed a defiitio of the discrete cetroaffie curves i R 2 ad R 3 Defiitio 28 A discrete plaar curve r : I R 2 is called a cetroaffie plaar curve if the edge taget vector t k is ot parallel to positio vectors r(k) ad r(k +), ad A discrete curve r : I R 3 is called a cetroaffie curve if the edge taget vectors t k, t k ad the positio vector r(k) are ot coplaar 3 Discrete plaar curves uder the affie trasformatio I this sectio we wat to cosider two ivariats ad their geometrical properties uder the affie trasformatio, although we call them the first ad secod cetroaffie curvatures just for uity, that because i the ext sectio we will use them together with cetroaffie torsios for a space discrete cetroaffie curve, oly uder the cetroaffie trasformatio Now let vectorvalued fuctio r : I Z R 2 represet a discrete curve C Below we give the defiitio of the cetroaffie curvatures by usig the otatios of the previous sectio Defiitio 3 For the discrete plaar curve C, if [ t k, t k ] = 0, we call its first cetroaffie curvature κ k = 0 at poit r(k), which implies the curve is a straight lie locally to r(k), where [ ] deotes the stadard determiat i R 2 If [ t k, t k ] 0, the first ad secod cetroaffie curvatures at the poit r(k) are defied by (3) κ k = [ t k, t k+ ] [ t k, t k ], κ k = [ t k, t k+ ] [ t k, t k ] 5,

6 Uder a affie trasformatio r = A r + b, we have t k = A t k So [ t k, t k+ ] [ t k, t k ] = (deta)[ t k, t k+] (deta)[ t k, t k ] = [ t k, t k+ ] [ t k, t k ], [ t k, t k+ ] [ t k, t k ] = (deta)[ t k, t k+ ] (deta)[ t k, t k ] = [ t k, t k+ ], [ t k, t k ] which implies κ k ad κ k are affie ivariats, of course, ad also cetroaffie ivariats Figure : The sig of the first curvature Now let us explai the geometrical meaig of the cetroaffie curvatures I Figure, P k deotes the ed poit of vector r(k) Therefore, the determiat [ t k, t k ] i R 2 exactly represets twice the oriet area of the triagle Pk P k P k+, ad [ t k, t k+ ] represets twice the oriet area of the triagle Pk P k+ P k+2 As a result, if the poits P k+2 ad P k lie o the same side of the straight lie P k P k+, the first cetroaffie curvature κ k takes positive value, ad if they lie o differet sides of the straight lie P k P k+, the first cetroaffie curvature κ k takes egative value I details, we ca see the left oe i Figure 2 Sice the triagles Pk P k P k+ ad Pk P k+ P k+2 have a edge P k P k+ i commo, accordig to the relatio of height ad area of a triagle, if poit P k+2 lies o differet lies, the cetroaffie curvature κ k is differet, ad if it lies o the same lie, the cetroaffie curvature κ k is same I the right of Figure 2, let P k+ Q = t k, P k+ R = t k Thus, [t k,t k ] represets the oriet area of the triagle Pk+ QR, ad [ t k, t k+ ] represets twice the oriet area of the triagle Pk+ QP k+2 These two triagles have a commo edge P k+ Q Similarly, by Eq (3), we ca also obtai the differet secod cetroaffie curvatures o the differet lies Figure 2: Geometric meaig of the curvatures Hece, we ca make use of above coclusios to decide the positio of the poit P k+2 I Figure 3, whe the poit P k+2 lies at differet itersectios of two group parallel lies, the cetroaffie curvature pair {κ k, κ k } is differet O the cotrary, give a pair {κ k, κ k }, the poit 6

7 P k+2 ca be uiquely decided, which lies at the itersectio of two straight lies κ k, κ k For example, i Figure 3, at the itersectio of two straight lies κ k = 2, κ k =, the poit P k+2 is uiquely determied Figure 3: Decidig a curve by the curvatures κ k ad κ k I fact, from Eq (3), we ca obtai the chai structure (32) r k+2 r k+ = κ k ( r k r k )+ κ k ( r k+ r k ) This shows that (33) r k+2 = κ k r k +( κ k κ k ) r k +(+ κ k ) r k+ Usig a simple matrix multiplicatio, it is coveiet to express Eqs (32) ad (33) by ( ) κk (34) ( r k+2 r k+, r k+ r k ) = ( r k+ r k, r k r k ) κ k 0 ad (35) ( r k+2, r k+, r k ) = ( r k+, r k, r k ) + κ k 0 κ k κ k 0 κ k 0 0 Exactly, this expressio ca be cosidered as a state trasitio process similar to a Markov chai, ad we otice that the sum of every colum of the trasitio matrix is If κ k 0, the matrices ( ) κk κ k 0 ad + κ k 0 κ k κ k 0 κ k 0 0 are reversible The from Eqs (34) ad (35) it follows that 0 (36) ( r k+ r k, r k r k ) = ( r k+2 r k+, r k+ r k ) κ k κ k 7 κ k

8 ad 0 0 (37) ( r k+, r k, r k ) = ( r k+2, r k+, r k ) 0 κ k 0 + κ k κ k κ k + κ k κ k, which are the iverse chais of Eqs (34) ad (35) I fact, the sum of every colum of these two trasitio matrices also is We shall ow start a discussio about two discrete plaar curves with same cetroaffie curvatures uder a affie trasformatio or a cetroaffie trasformatio, ad fid how the cetroaffie curvatures affect a discrete plaar curve by the chai structure (35) The followig two propositios tell us that two discrete plaar curves with same cetroaffie curvatures are affie equivalet, ad they are cetroaffie equivalet up to a traslatio trasformatio Propositio 32 Give two sequeces of umber {κ,κ 2, } ad { κ, κ 2, }, where κ k 0, κ k 0, k Z, up to a cetroaffie trasformatio, there exists a discrete plaar curve r : I R 2 such that κ k ad κ k is the first ad secod cetroaffie curvature of the curve r Proof I a plae, give ay two liearly idepedet vector groups { t 0, t } ad { e 0, e }, there must exists a ivertible matrix A of size 2 such that ( e 0, e ) = A( t 0, t ) Sice the cetroaffie curvatures are ivariat uder a affie trasformatio, of course, they also are ivariat uder a cetroaffie trasformatio, we ca choose two fixed liearly idepedet vectors { e 0, e } as the first two taget vectors { t 0 = r() r(0), t = r(2) r()} From Eq (32) we ca obtai e 2, e 3, i tur The we make e 0, e, head ad tail dockig, which is a discrete curve with cetroaffie curvatures κ k ad κ k,(k = 0,,2, ) The above propositio tells us a discrete plaar curve ca be determied by cetroaffie curvatures with respect to a cetroaffie trasformatio I fact, with differet startig poit of e 0, the curve is differet uder the cetroaffie trasformatio I the followig propositio, we kow they are affie equivalet Propositio 33 Assume two discrete plaar curves r(k), r(k) have same cetroaffie curvatures o the correspodig poits, the, there exist a o-degeerate matrix A of size 2 ad a costat vector C such that r(k) = A r(k)+ C, for all k Z, that is, these two curves are affie equivalet Proof Clearly, there is a o-degeerate matrix A of size 2 satisfyig that From Eq (32), we get Oe after aother, it follows that { r() r(0), r(2) r()} = A { r() r(0), r(2) r() } r(3) r(2) = A( r(3) r(2)) r(k +) r(k) = A( r(k +) r(k)),k Z 8

9 A costat vector C is give by C = r(0) A r(0), It is easily see that r(0) = A r(0)+ C, r() = r(0) + r() r(0) = A r(0)+ C +A( r() r(0)) = A r()+ C, r(2) = r() + r(2) r() = A r()+ C +A( r(2) r()) = A r(2)+ C, Clearly, we obtai r(k) = A r(k)+ C, k Z, which completes the proof of the propositio Remark 34 Sice the cetroaffie trasformatio does ot iclude the traslatio trasformatio, the differet choice of the startig poit will geerate differet discrete cetroaffie curves However, with give cetroaffie curvatures, there exists oe ad oly oe curve uder affie trasformatio, that is, two discrete plaar curves with same cetroaffie curvatures are affie equivalet If two discrete plaar curves r(k), r(k) are cetroaffie equivalet, that is, there exists a cetroaffie trasformatio A such that r(k) = A r(k), k Z, these two curves have same cetroaffie curvatures o the correspodig poits O the other had, if we fix a iitial vector r 0, with give cetroaffie curvatures, there exists oly oe discrete plaar curve I the followig, we wat to describe some results that belog to the global differetial geometry of a discrete plaar curve Observe Eq (35), we obtai (38) ( r p+2, r p+, r p ) = ( r 2, r, r 0 ) + κ 0 κ κ p 0 κ κ p 0 κ p κ p 0 κ p 0 0, p Z For a discrete closed curve with period p Z, we have ( r p+2, r p+, r p ) = ( r 2, r, r 0 ) Thus the followig lemma is obvious Lemma 35 A discrete cetroaffie curve is a closed curve with period p Z if ad oly if + κ 0 κ κ 0 κ 0 0 where E is the idetity matrix of size 3 + κ 2 0 κ 2 κ 2 0 κ κ p 0 κ p κ p 0 κ p 0 0 = E,

10 Notice that immediately, we have det + κ k 0 κ k κ k 0 κ k 0 0 = κ k, Corollary 36 If a discrete plaar curve is closed with period p, its first cetroaffie curvature satisfies that κ κ 2 κ p = As we kow, i classical differetial geometry, a closed plaar curve with costat curvature is a circle Naturally, it is iterestig to cosider the similar problems for a discrete plaar curve Firstly, let us give the followig defiitio Defiitio 37 A discrete cetroaffie curve is called costat curvature cetroaffie curve if its first ad secod cetroaffie curvatures are costat I the particular case, accordig to the defiitio of the cetroaffie curvature, a discrete plaar curve withκ = 0 isalie Hece, fromowowe assume κ 0 The followig propositioshows how to get a discrete closed plaar curve depedig o the costat cetroaffie curvatures Propositio 38 A discrete plaar curve with costat cetroaffie curvatures is closed if ad oly if the curvatures κ =, κ 2, ad there exist θ R,p,l Z, such that cosθ = κ, pθ = 2lπ, 2 where p is the period of the discrete closed curve, p ad l are coprime Proof It is easy to see from Lemma 35 ad Corollary 36, that a discrete plaar curve with costat cetroaffie curvatures κ ad κ is closed if ad oly if there exists a iteger p Z such that (39) To take the fact det ito accout, easily we get κ = or κ = + κ 0 κ κ 0 κ κ 0 κ κ 0 κ 0 0 p = E = κ Now if κ =, the trasitio matrix ca be rewritte as + κ 0 κ By a simple calculatio, we obtai its eigevalues λ =,λ 2,3 = κ κ2 2 ± +4 Sice λ 2, λ 3, it is 2 p + κ 0 impossible to fid a iteger p satisfyig that κ 0 = E κ 0 O the other had, if κ =, the trasitio matrix is κ 0, ad its eigevalues 0 0 are λ =,λ 2,3 = κ κ2 2 ± 4 Accordig to Eq (39) we obtai λ = λ 2 = λ 3 =, which 2 0

11 shows that κ 2, at the same time, there is a θ satisfyig that κ κ2 2 ± 4 = cosθ± siθ 2 AgaifromEq (39), wekow(cosθ± siθ) p = exp(± pθ) =, whichimplies pθ = 2lπ, where l Z, p ad l are coprime The the trasitio equatio (35) tells us this is a closed discrete plaar curve with period p O the cotrary, if these coditios hold, Eq (39) is satisfied Together with Eq (35), we ca obtai a discrete closed plaar curve I the classical differetial geometry, it is well kow that a closed regular plaar simple curve is covex if ad oly if its curvature is either always o-egative or always o-positive, ie, if ad oly if the turig agle (the agle of the taget to the curve) is a weakly mootoe fuctio of the parametrizatio of the curve Similarly, for a discrete plaar curve we obtai Propositio 39 A discrete closed plaar simple curve C is covex if ad oly if its first cetroaffie curvature κ k > 0 Proof I fact, a discrete closed plaar simple curve is a plaar polygo Exactly, covexity ad cetroaffie curvatures are ivariat uder the affie trasformatio i the plae Therefore, we ca cosider the problem by affiely trasformig the polygo to a fixed polygo o the Euclidea plae As we kow, a polygo is covex if ad oly if each of its iterior agles has a measure that is strictly less tha π From Eq (3), we obtai κ k = t k+ siα k+ t k siα k, where α k is the k th iterior agle of the polygo Sice 0 < α i < π(i =,2,,p), we get every κ k > 0 O the other had, if κ i > 0(i =,2,,p), we obtai siα,siα 2,,siα p have the same sig Hece, 0 < α i < π(i =,2,,p), which implies the polygo is covex By a simple calculatio, the followig result is obvious Remark 30 If a discrete closed curve is a triagle, the its cetroaffie curvatures satisfy that κ = κ 2 = κ 3 =, κ = κ 2 = κ 3 = If a discrete closed curve is a parallelogram, we have κ = κ 2 = κ 3 = κ 4 =, κ = κ 2 = κ 3 = κ 4 = 0 If a covex polygo has p sides, the its iterior agle sum is give by the equatio (p 2)π Obviously, usig the same method as above ad the graph show i the right of Figure 2, it is easy to prove the followig result Corollary 3 If a p polygo except parallelogram is covex, where p > 3, its secod cetroaffie curvature κ k >, ad there are o more tha 2 o positive secod cetroaffie curvatures κ Furthermore, if κ k 0 ad κ l 0, we must have k l The followig two propositios illustrate how to get a covex closed curve depedig o costat cetroaffie curvatures ad estimate whether it has self-itersectios Propositio 32 A discrete plaar curve with costat cetroaffie curvatures is the simple covex closed curve if there exist θ ad p, where θ R,p Z, satisfyig that cosθ = κ, pθ = 2π 2 ad κ =

12 Proof From Propositio 38, the curve is a closed curve with period p ad p 3 Obviously, we have κ < 2 Sice cosθ = κ, pθ = 2π, p 3, we have κ = or 0 κ < 2 If κ =, it 2 follows that θ = 2π,p = 3, which is a triagle Certaily this is a simple covex closed curve 3 If κ = 0, we obtai θ = π,p = 4 By a simple calculatio, we get κ =, κ = 0 for the 2 square show i Figure 4, which is a simple covex closed curve Accordig to Propositio 33 ad Theorem 2, it is clearly that a discrete plaar curve with κ =, κ = 0 is a simple covex closed curve with period 4 Figure 4: Simple covex closed curves If 0 < κ < 2 ad there exist θ ad p, such that cosθ = κ, pθ = 2π, we make a equilateral 2 polygo of size p as i Figure 4, where poit A is the ceter of the equilateral polygo, P i are the ed poits of vector r(i),i = 0,,, It is easy to check that the first cetroaffie curvature κ = Now let us calculate its secod cetroaffie curvature κ c By usig the otatio T = ( cosθ siθ siθ cosθ ), we have AP k = AP k T, AP k+ = AP k T 2, 2 AP k+2 = AP k T 3

13 The κ c = [ t k, t k+ ] [ t k, t k ] = [ AP k AP k, AP k+2 AP k+ ] [ AP k AP k, AP k+ AP k ] = [(T E) AP k,t 2 (T E) AP k ] [(T E) AP k,t(t E) AP k ] = [ AP k,t 2 AP k ] [ AP k,tap k ] = [ AP k, AP k+ ] [ AP k, AP k ] = Area( AP k P k+ ) Area( APk P k ) = 2cosθ = κ So this equilateral polygo is a simple covex closed curve with costat cetroaffie curvature κ = ad κ Agai from Propositio 33 ad Theorem 2, a discrete plaar curve with costat cetroaffie curvatures κ = ad above κ is a simple covex closed curve Figure 5: Closed curves with self-itersectios Propositio 33 A discrete plaar curve with costat cetroaffie curvatures is closed curve with self-itersectios if κ =, κ 2, θ R,p,l Z, such that cosθ = κ ad pθ = 2lπ,l >, 2 where p ad l are relatively prime Proof Usig the similar methods as Propositio 32, we ca obtai the cetroaffie curvatures of the equilateral polygo of size p show i Figure 5 By a direct calculatio, we obtai κ =, κ = 2 cos θ Clearly, this closed curve has self-itersectios From Propositio 33 ad Theorem 2, a discrete plaar curve with costat cetroaffie curvature κ = ad above κ is a closed 3

14 curve with self-itersectios Accordig to the proofs of Propositio 32 ad Propositio 33, we kow if the period p is eve, A is the symmetric ceter of the curve Sice the partitio ratio of 3 poits is ivariat uder affie mappigs, it is immediate to get Corollary 34 A discrete plae closed curve with costat cetroaffie curvatures is cetrosymmetric if ad oly if its period is eve Fially, we ca defie the affiely regular polygo usig the affie curvatures Defiitio 35 A plaar polygo is a affiely regular polygo with period p if ad oly if it have costat affie curvatures κ =, κ = 2cos 2lπ 2l, where p ad l are relatively prime ad < p p Especially, l =, it is a affiely regular simple polygos (a simple polygo is oe that does ot itersect itself aywhere) 4 Discrete cetroaffie space curve i R 3 Let curve r : I Z R 3 be a cetroaffie discrete curve deoted by C, ad the by the defiitio 28, we have [ r k, t k, t k ] 0, where [ ] deotes the stadard determiat i R 3 I the followig the cetroaffie curvatures ad cetroaffie torsios of a cetroaffie discrete space curve i R 3 will be defied Defiitio 4 The first, secod cetroaffie curvatures ad cetroaffie torsios of the discrete cetroaffie curve r at poit r(k) are defied by (4) κ k := [ r k+, t k, t k+ ] [ r k, t k, t k ], κ k := [ r k+, t k, t k+ ], τ k := [ t k, t k, t k+ ] [ r k, t k, t k ] [ r k, t k, t k ] ByDefiitio4,uderacetroaffietrasformatioR 3 x A x R 3,whereA GL(3,R),it is easy to see that the first, secod cetroaffie curvatures ad cetroaffie torsios are ivariat However, uder a affie trasformatio x A x + C, where C R 3 is a costat vector, the first, secod cetroaffie curvatures ad cetroaffie torsios may chage Hece, we have Propositio 42 The first, secod cetroaffie curvatures ad cetroaffie torsios are cetroaffie ivariats ad ot affie ivariats I fact, [ r k, t k, t k ] = [ r k, r k, r k+ ] The the cetroaffie curvatures ad torsios ca be rewritte as (42) κ k = [ r k, r k+, r k+2 ] [ r k, r k, r k+ ], κ k = [ r k+, t k, r k+2 ] [ r k, r k, r k+ ], τ k = [ t k, t k, t k+ ] [ r k, r k, r k+ ] By a direct calculatio, it follows that (43) r k+2 = κ k r k +( κ k κ k ) r k +(τ k + κ k +) r k+, k Z, ad (44) ( r k+2, r k+, r k ) = ( r k+, r k, r k ) τ k ++ κ k 0 κ k κ k 0 κ k 0 0, 4

15 which are the three dimesioal curve chai structures This formula is called the Freet-Serret formula of a discrete cetroaffie curve O the other had, whe τ = 0, from Eq (43) we get the chai of edge taget vector (45) t k+ = κ k t k + κ k t k, k Z, which is coicidet with Eq (32) If κ k 0, we otice that the iverse chai ca be represeted as (46) ( r k+, r k, r k ) = ( r k+2, r k+, r k ) 0 0 κ k 0 τ k++ κ k κ k 0 + κ k κ k I this sectio, we oly cosider the discrete cetroaffie space curve C uder the cetroaffie trasformatio Firstly, the followig propositio states that with the give first, secod cetroaffie curvatures ad cetroaffie torsios, the curve is oly determied up to a cetroaffie trasformatio Propositio 43 Two curve C ad C are cetroaffie equivalet if ad oly if they have same cetroaffie curvatures κ k, κ k ad torsios τ k, for all k I Z Proof It is easy to see from Eq (4) that if the curves C ad C satisfy that r(k) = A r(k), where A is a 3 3 matrix, they have same curvatures κ k, κ k ad torsios τ k O the other had, if curves C ad C have same cetroaffie curvatures ad torsios at correspodig poits, we eed to show they are cetroaffie equivalet Obviously there exist a matrix A of size 3 such that ( r(0), r(), r(2)) = A( r(0), r(), r(2)) From Eq (43), by the same cetroaffie curvatures κ k, κ k ad torsios τ k, it is simple to prove that r(k) = A r(k), k I Z This meas the curves C ad C are cetroaffie equivalet Next, we will cosider the geometric iterpretatio for the cetroaffie curvatures ad cetroaffie torsios by figures We deote the ed poit of vector r(k) by P k ad the plaar icludig the poits P k,p k ad P k+ by π 0 I Figure 6, accordig to Eq (4) we kow if the poit P k+2 lies differet plaes which parallel to the plae π 0, the torsios τ k are differet If the poit P k+2 lies differet place i a same plae which parallels to the plae π 0, the torsios τ k are same Similarly, i the left of Figure 7, let π represet the plae cotaiig the poits P k,p k+ ad the origi O By Eq (4) we obtai if P k+2 lies differet plaes which parallel to the plae π, the curvatures κ k are differet If P k+2 lies differet place i a same plae which parallels to the plae π, the cetroaffie curvatures κ k are same I the right of Figure 7, we give a vector P k+ P k = P k P k The [ OP k, OP k, OP k+ ] = [ OP k, OP k+, OP k ] Let π 2 represet the plae cotaiig the poits P k,p k+ ad the origi O Agai from Eq (4), we coclude that if P k+2 lies differet plaes which parallel to the plae π 2, the curvatures κ k are differet If P k+2 lies differet place i a same plae which parallels to the plae π 2, the cetroaffie curvatures κ k are same 5

16 Figure 6: Cetroaffie torsios i differet plaes Figure 7: Cetroaffie curvatures i differet plaes 6

17 From the defiitio ad above geometric iterpretatio, we kow if τ k = 0, the curve is a plaar curve show i Figure 8 I this case, by compariso, we ca fid the defiitios of cetroaffie curvatures are coicidet with the plae situatio i the above sectio I Figure 8, let P k+2 P = P k P k Because the poits P k,p k,p k+,p k+2 are i the same plae π, so is the poit P Hece, the first cetroaffie curvature κ k = [ r k+, t k, t k+ ] is the ratio of the oriet area [ r k, t k, t k ] of the triagle Pk P k+ P k+2 ad the oriet area of the triagle Pk P k P k+ At the same time the secod cetroaffie curvature κ k = [ r k+, t k, t k+ ] is the ratio of the oriet area of the triagle [ r k, t k, t k ] Pk+ P k+2 P ad the oriet area of the triagle P k P k P k+ Exactly, their geometry meaig is same as a plaar curve defied i the above sectio Therefore we have Figure 8: Cetroaffie plaar curve i R 3 Remark 44 If τ k = 0, k Z, the curve C is a plaar curve ad the cetroaffie curvatures i Eq (4) are as same as defied i Eq (3) for the plaar curve A discrete space curve with period p is closed if ad oly if r(k +p) = r(k), k Z Through the three dimesioal curve chai, that is, Eq (44), it is easy to see Lemma 45 A discrete cetroaffie space curve is a closed curve with period p if ad oly if the cetroaffie curvatures ad torsios satisfy that + κ +τ 0 + κ 2 +τ κ p +τ p 0 κ κ 0 κ 2 κ 2 0 κ p κ p 0 = E, κ 0 0 κ κ p 0 0 where E is the idetity matrix of size 3 Observe that It is immediate to obtai det + κ k +τ k 0 κ k κ k 0 κ k 0 0 = κ k Corollary 46 If a discrete cetroaffie curve is closed with period p, the κ κ 2 κ p = I the above sectio, we have cosidered a discrete plaar curve with costat cetroaffie curvatures Similarly, we ca obtai 7

18 Propositio 47 Let the curve C be a discrete cetroaffie closed p polygo with costat cetroaffie curvatures κ, κ ad torsios τ, the we have κ = or κ = Moreover, if κ =, the curve is plaar curve If κ =, we have 0 < κ < 4,τ < 0, τ = 2 κ, ad there exist a real umber θ ad a iteger l which is relatively prime to p satisfyig that cosθ = κ,pθ = 2lπ, 2 where p is eve umber Proof Sice C is a discrete cetroaffie closed curve with costat cetroaffie curvatures κ, κ ad torsios τ, we have from Corollary 46 that κ = or κ = Obviously, if κ =, p is eve umber + κ+τ 0 Assume λ,λ 2,λ 3 are the eigevalues of the matrix κ κ 0 The eigevalue κ 0 0 equatio of this matrix is (47) λ 3 (τ + κ+)λ 2 +(κ+ κ)λ k = 0 Hece we obtai (48) λ +λ 2 +λ 3 = τ + κ+, λ λ 2 +λ 2 λ 3 +λ 3 λ = κ+ κ, λ λ 2 λ 3 = κ p + κ+τ 0 That the curve C is closed, that is, κ κ 0 = E, implies λ,λ 2,λ 3 are ot equal κ 0 0 to each other ad (49) λ p = λ p 2 = λ p 3 = If κ =, from Eqs (48) ad (49) we ca assume λ =, λ 2 = cosθ+ siθ, λ 3 = cosθ siθ, where pθ = 2lπ, p, l are relatively prime Immediately Eq (48) geerates λ 2 +λ 3 = τ + κ, λ 2 +λ 3 = κ, which implies τ = 0 ad the curve C is a plaar curve If κ =, from Eqs (48) ad (49) we ca assume λ =, λ 2 = cosθ+ siθ, λ 3 = cosθ siθ, where pθ = 2lπ ad p,l are relatively prime Agai from Eq (48) we get Fially we obtai λ 2 +λ 3 = τ + κ+2, λ 2 +λ 3 = 2 κ τ = 2 κ, cosθ = κ 2 The we complete the proof From the above proof, it is clear that if κ =, r(k + p ) = r(k), which implies the curve is 2 cetrosymmetric ad the ceter of symmetry is the origi O Hece, we have Corollary 48 If a discrete cetroaffie closed curve with costat cetroaffie curvatures ad torsios is ot a plaar curve, it is symmetric aroud the origi O Corollary 49 If a discrete cetroaffie curve with costat cetroaffie curvatures ad torsios is closed with period p, where p is a eve umber, the it is cetrosymmetric 8

19 5 Flows o curves Curve-shorteig flowisthesimplest exampleofacurvatureflow Itmoves eachpoitoaplaar curve γ i the iwards ormal directio ν with speed proportioal to the siged curvature κ at that poit, as described by the equatio γ = κ ν The ame curve-shorteig comes from t the fact that the curve is always movig so as to decrease its legth as efficietly as possible I this sectio, the geeral form of the flow o curves is cosidered Furthermore, it ca be exteded to the discrete curves See [7] for more details 5 Flows o smooth curves The motio of a curve r : I R N i space could be described by applyig some vector field v I geeral v might deped o the whole curve Exactly, if v depeds oly o a small eighborhood at each poit of the curve, we call the geerated flow a local flow With these coditios, the evolutio process of r uder the flow geerated by v ca be described by a differetial equatio (5) t r = v( r, r, r, ) A oe-parameter family of curves (52) r : I J R N which is a solutio of Eq (5) i the sese that (53) t r(s,t) = v( r(s,t), r (s,t), r (s,t), ) =: v(s,t) for all (s,t) I J is called the evolutio of the curve r 0 (s) = r(s,0) uder the flow give by v For this particular iitial curve r 0, the vector field v becomes a o-parameter family of vector fields alog the parametrizatio (54) v : I J R N, ad Eq (5) becomes (55) t r(s,t) = v(s,t) The map (56) Φ : (t, r) Φ t r, where Φ t r(s) = r(s,t) is the evolutio of r is called the curve flow give by v Note that i geeral Φ might ot be well defied due to lack of existece ad uiqueess of solutios of Eq (5) for arbitrary r 0 Additioally if oe might wat the flow to be geometric, ie oly deped o the shape of the curve It should be ivariat with respect to Euclidea motios, reparametrizatio of the curve The flow is the well defied o the correspodig equivalece classes of parametrized curves Example(plaar geometric flow) For plaar curves these two coditios ca be realized by the asatz: v = v(κ,κ,κ, ) = α(κ,κ,κ, ) T +β(κ,κ,κ, ) N Exactly, the flow ca be geeralized i affie space if the flow is ivariat with respect to affie trasformatio 9

20 52 Flows o discrete curves The previous sectio was devoted to the cotiuous case We ow study the discrete case For I := [0,,] Z fiite iterval, I = Z := Z/Z ad I = Z, we defie the space (57) C I := { r : I R N } of fiite,fiite closed ad ifiite curves respectively A flow of discrete curves is give by a vector field (58) v : C I TC I, r v [ r ] T r C I, or o some submaifold U C I I the fiite case we have T r C I = (R N ) So v gives a directio i R N at every vertex k which possibly depeds o the whole curve r We state this relatio as (59) v k [ r ] R N For a give iitial curve r : I R N the vector field v o C I becomes a oe-parameter family of vector fields alog the parametrizatio of the curve (50) v : I J R N, where J R is a ope iterval The actio of the flow leads to a cotiuous deformatio of the curve r k = r k (0), (5) r : I J R N satisfyig that (52) t r(t) = v k (t) By a local flow we mea a flow which at every vertex k oly depeds o the curve at the adjacet vertices, ie r k, r k, r k+ : (53) v k [ r ] = v( r k, r k, r k+ ) I affie geometry, the result of a flow is very differet A tiy curve segmet ca be affiely equivalet to a huge oe So we defie its stability depedig o its affie ivariats, such as cetraffie curvatures ad torsios Defiitio 5 A discrete curve is stable if it remais its cetroaffie curvatures ad the torsios uchaged i the ext descedat A discrete curve is periodically stable if oe of its descedats have the same cetroaffie curvatures ad torsios as that 6 Trasversal flow o discrete cetroaffie space curves Now we exted the flow to a cetroaffie space curve, which implies the flow is ivariat with respect to the affie trasformatio or cetroaffie trasformatio As metioed previously, the cetroaffie curvatures are affie ivariat whe the discrete curve lies i 2-dimesioal plae Whe the curve is plaar, the results are suitable for the affie trasformatio Sice the cetroaffie curvatures are coicidet for a discrete plaar curve o matter is i 2-dimesioal plae or i 3-dimesioal space, here we cosider flows o the discrete curve i 3-dimesioal 20

21 space We draw a aalogy betwee the two cases by replacig a curve with a discrete curve, ad the time with a discrete time For a cetroaffie space curve, its positio vector field is always trasversal to the osculatio plae The discrete trasversal flow ca be defied by (6) t r k = v k := α k r k, where α k = α k [ r ] should deped o the curve i ay way ad be cetroaffiely ivariat Let r m R3 be a discrete cetroaffie curve, where is the idex of the vertices ad m is the discrete deformatio parameter For a local discrete trasversal flow r m, we have (62) r m+ r m = αm r m, where α m is cetroaffiely ivariat This equatio ca also be writte as (63) r m+ = (+α m ) r m By takig the otatio (64) β m = +α m, the trasfer equatio is easily show (65) ( r m+ m+ +, r, r m+ ) = ( r m +, r m, r m )Mm, β m where M m = β m β m I fact, we have obtaied the structure equatio (35) for a discrete cetroaffie space curve i the previous sectio By a simplificatio, the followig otatio is used (66) L m = + κ m +τm 0 κ m κm 0 κ m 0 0, (L m ) = 0 0 κ m 0 + κm +τm 0 κ m κ m + κ m κ m Now, by usig the compatibility coditio of the liear system (35) ad (65) it is o difficult to get L m M m + = M m L m+, (67) (68) (69) β+2 m (+ κm +τm ) = βm + (+ κm+ +τ m+ ), β m +2(κ m + κ m ) = β m (κ m+ + κ m+ ), β m +2 κm = β m κm+ 2

22 Obviously, if β m m+ = 0, by Eq (63), we have r = 0, which is cotrary to the defiitio of the discrete cetroaffie curve So i the followig we should assume β 0 By a direct computatio, from Eqs (67)-(69), it shows the cetroaffie curvatures ad torsios of ext geeratio vertex r m+ are (60) (6) τ m+ = βm +2 β m κ m+ = βm +2 β m κ m + βm +2 β m + κ m, (+ κ m +τ m ) βm +2 (κ m β m + κ m ), (62) κ m+ = βm +2 β+ m Therefore, we may coclude that (κ m + κ m ) βm +2 κ m β m Propositio 6 Uder a discrete trasversal motio, the curve r m ad r m+ have the relatio (60)-(62) I order to be more clear, we could chage Eqs (60)-(62) to a matrix form (63) τ m+ κ m+ κ m+ = β m +2 β m + 0 β+2 m βm +2 β+ m β m β m +2 β m β+2 m βm +2 β m β m β+2 m βm β+ m +2 β m β m +2 β m β m +2 β m τ m κ m κ m Now, the followig propositio shows that if β is costat, the trasversal flow of a discrete curve would keep stable Propositio 62 If β is costat, the the discrete trasversal motio r m+ of a discrete curve r m is cetroaffiely equivalet to the curve r m Whe we assume κ m +κ m 0, β is costat if ad oly if the discrete trasversal motio r m+ of a discrete curve r m is cetroaffiely equivalet to the curve r m Proof If β is costat, from Eq (63) it is easy to get τ m+ = τ m, κ m+ = κ m,κ m+ = κ m, which implies the curve r m+ is cetroaffie equivalet to the curve r m O the other had, if the curve r m+ is cetroaffie equivalet to the curve r m, the τ m+ = τ m, κ m+ = κ m,κ m+ = κ m From Eq (6), we get β m = β+2 m The by Eq (62) ad κ m +κ m 0, we have β+ m = βm +2 Agai from Eq (60) we get βm = βm + Hece, we obtai β m is costat The by observig Eq (60), we ca state the followig propositio 22

23 Propositio 63 If the discrete trasversal motio r m+ of a discrete plaar curve r m is still plaar, that is, τ m+ = τ m = 0, the coefficiets of motio satisfy that (64) or (65) β m +2 β m +2 = = ( β m β m + β m + (κ m + κ m )+, β m, = κ m ( β m β m + β m κ m + (+ κ m β ) + m )(+ κ m, κm κm,κm )Tra, β m ) κ m ( β m ) β m Especially, they ca be represeted as the trasitio trasformatio + κ m 0 (66) (,, ) = (,, ) κ m β+2 m β+ m β m β+ m β m β m κm 0 κ m 0 0 ad (67) ( κ m+ κ m+ ) = β m +2 β m + 0 β+2 m βm +2 β+ m β m β+2 m β m ( κ m κ m ) Furthermore, accordig to Eq (64), the followig propositio is obvious Propositio 64 For a plaar p polygo, if its ext geeratio is still plaar uder the trasversal motio, the coefficiets of motio β m,β m 2,,β m p should satisfy that (68) S β m β m 2 β m 3 β m p = 0, where S = κ m κm + κ m 0 0 κ m κ m 2 κ m 2 κm 2 + κ m κ m 3 κ m 3 κ m 3 + κ m κ m 4 κ m 4 κm κ m p κ m p + κ m p + κ m p 0 0 κ m p κ m p κ m p I order to obtai the solutios of the above equatio, firstly, we eed to study the rak of coefficiet matrix S Propositio 65 The rak of matrix S is p 3 23

24 Proof Whe τ = 0, from Eq (44), we have S ( r m, r m 2,, r m p ) Tra = 0, which implies the colum vectors of ( r m, r m 2,, r m p )Tra are three solutios of liear equatio Sx = 0 Sice r m k, r m k+, r m k+,(k =,2,,p 2) are liearly idepedet, it is easy to see that the rak of ( r m, r m 2,, r ) m p is 3 If there exists a vector group ( r, r 2,, r p ) satisfyig that S ( ) Tra r, r 2,, r p = 0 From Propositio 43, we kow there must be a matrix A of size 3 satisfyig that r k = A r m k (k =,2,,p) By the traspositio, it follows that ( r, r 2,, r p ) Tra = ( r m, r m 2,, r m p ) TraA Tra, which implies arbitrary solutio of liear equatio Sx = 0 ca be liearly represeted by colum vectors of ( r m, r m 2,, r m p ) Tra If we assume the colum vectors of ( r m, r m 2,, r m p ) Tra are V,V 2,V 3, the vector group {V,V 2,V 3 } is a system of fudametal solutio for liear equatio Sx = 0 Hece, it is clear that the rak of matrix S is p 3 Now from Eq (68) ad the proof of Propositio 65, we have β m β m 2 β m 3 β m p = a V +a 2 V 2 +a 3 V 3, where a,a 2,a 3 are arbitrary costat Ifact, β m should becetroaffie ivariat, but V,V 2,V 3 areot So we eed to fidasystem of fudametal solutios for liear equatio Sx = 0, which are cetroaffie ivariat As we kow, there exists a matrix A, such that A r m = (,0,0)Tra,A r m 2 = (0,,0)Tra,A r m 3 = (0,0,)Tra Hece, we obtai a stadard vector group r m i = A r m i,(i =,2,,p) which are ivariat uder cetroaffie trasformatio Now assume the colum vectors of ( r m, r m 2,, r m p ) Tra are V, V 2, V 3, ad the vector group { V, V 2, V 3 } is a system of fudametal solutios for liear equatio Sx = 0 ad cetraffie ivariat Thus we obtai Corollary 66 If a trasversal motio of a plaar p polygo r m remais plaar, the coefficiets β m satisfy that β m β m 2 β m 3 β m p = a m V +a m V 2 2 +a m V 3 3, where a m,a m 2,a m 3 are some cetroaffie ivariats that ca esure β m i 0(i =,2,,p) 24

25 Hece, if a m = am 2 = am 3, we have βm = β2 m = = βp m = a m Accordig to Propositio 62, this implies the discrete curve r m is stable uder the above trasversal motio Computer experimets show that there are some plaar polygos whose trasversal flows will reach stable I the followig, we give a example, which shows after multistep iteratio the coefficiets a m,am 2,am 3 will be same Example of stable trasversal flow Let us take a m = p i= κm i p, a m 2 = p i= κm i p, a m 3 = am +am 2 2 It is obvious that a m,a m 2,a m 3 are cetroaffie ivariat Usig computer experimets, we choose a covex plaar petago, which will keep plaar durig the iteratio process I Table, we ca see that, after the thirty-eighth step, this trasversal cetroaffie flow will reach stable That is, from the o, the mea value of the first cetroaffie curvatures is equal to that of the secod cetroaffie curvatures Table : A stable trasversal flow Iitial poits (0,22,) (8,2,) (2,0,) (37,2,) (48,28,) a a 2 κ κ κ κ κ κ κ κ Taget flow o discrete cetroaffie curves Sice taget vectors of a discrete cetroaffie curve r live o edges, it is ot istatly clear what the taget directio at a vertex k should be If we wat it to deped oly o the eighborig taget vectors, a obvious symmetric choice would be t k + t k However, if the motios of the discrete curve always lie o the taget plae, we ca defie the motios as the discrete taget flow of discrete cetroaffie curves, that is, (7) t r k = v k := α k t k +β k t k, where α k = α k [ r ] ad β k = β k [ r ] should deped o the curve i ay way ad be cetroaffie ivariat Now let r m R 3 be a discrete cetroaffie space curve, where is the idex of the vertices ad m is the discrete deformatio parameter For a local discrete taget flow r m, it is ature to see (72) r m+ r m = αm t m +β m t m = αm r m + +(βm αm +) r m βm r m, where α m ad β m are cetroaffie ivariat This equatio ca also be writte as (73) r m+ = ( r m +, r m, r m )(αm, αm +βm, βm )Tra 25

26 The by Eqs (35) ad (66) we get ad r m+ + = ( r m +2, r m +, r m )(α m +, α m + +β m +, β m +) Tra = ( r m +, r m, r m )Lm (αm +, αm + +βm +, βm + )Tra, r m+ = ( r m, r m, r m 2 )(αm, αm +βm, βm )Tra = ( r m +, r m, r m )(Lm ) (α m, αm +βm, βm )Tra Obviously, the three equatios above ca be regarded as state trasitio equatios (74) ( r m+ m+ +, r, r m+ ) = ( r m +, r m, r m )Mm, where M m = +β+ m +αm + (τm + κm ) αm βm κ m α+ m (κm + κm ) βm + β m αm + αm + βm κ m (+τ m + κm ) α m + κm β m α m βm κm κ m Similarly, the compatibility coditio of the liear system (35) ad (74) L m Mm + = Mm Lm+ yields (75) M m + κ m+ +τ m+ κ m+ κ m+ κ m+ = L m +α m +2(τ m + + κ m +)+β m +2 β m +2 αm +2 (κm + + κm + ) α m +2 κm + By Eq (74) it is easy to see M m is o-degeerate So it follows that + κ m+ +τ m+ (76) = (M m ) L m κ m+ κ m+ κ m+ More clearly, it ca be writte as (77) τ m+ κ m+ κ m+ Ideed, we have =α+2 m (M m ) L m (M m ) L m +α m +2 (τm + + κm + )+βm +2 β m +2 α m +2(κ m + + κ m +) α m +2 κm β m +2 β m +2 0 τ m + κ m + Remark 7 By the defiitio of discrete taget flow, it is easy to see the taget flows of a plaar discrete curve are still plaar O the other had, this result ca also be show by usig that the sum of elemets of every colum vector is i Eq (76) Hece, if τ m = 0, it is clear that τ m+ = κ m +

27 7 Iteratio of defiite proportioal divisio poit The midpoit map is perhaps the simplest polygo iteratio Startig with a N polygo P, we create a ew N polygo P 2 whose vertices are midpoits of the edges of P For almost every choice of P, if we iterate this process, the obtaied sequece of polygos P k will coverge to its cetroid Furthermore, Berlekamp et al cocluded that the descedats of a plaar polygo approach a affiely regular polygo The results ca be exteded to the more geeral trasformatio with the formula r m+ = a 0 r m +a r m + + +a N r m +N, where a 0,a,a N are ay costats ad where the N subscripts o the r m +k are to be computed modulo N[2] For o-plaar polygos, they obtaied almost all o-plaar polygos lack plaar descedats If the first descedat is o-plaar the so are all the rest O the other had, all o-plaar polygos have descedats which differ arbitrarily little (relative to their size) from plaar polygos Ad almost all o-plaar polygos have descedats which differ arbitrarily little (relative to their size) from plaar covex polygos[2] Now we ca visually show these results by its cetroaffie curvatures ad torsios By Defiitio35adRemark44, τ = 0represets theplaardiscrete curve, adκ=, κ = 2cos 2lπ N,τ = 0 represet the plaar affiely regular polygos To make the results above more clearly, we ca use the taget flow o a polygo with the related cetroaffie curvatures ad torsios Exactly, the iteratio of polygos with defiite proportioal divisio poit ca be described by r m+ = ( α) r m +α r m +, where α is costat ad 0 < α < Usig Eq (7), we have α m = α, βm = 0 By usig Eq (77), we get the iteratio of torsios ad curvatures for the taget flow r m, which may be writte as (78) (79) (70) τ m+ κ m+ κ m+ = (α α2 ) κ m +α2 κ m +( α)2 α(α ) α( α)2 τ m + +(α α 2 ) κ m + +α 2 κ m + +( α) 2 α( α) 2 τ m +(α α 2 ) κ m +α 2 κ m +( α) 2 α( α)τm + +(α α 2 ) κ m + +α 2 κ m + +( α) 2, α(α ) = ( κ m + α τ m α κm )α( α)2 + +(α α2 ) κ m + +α2 κ m + +( α)2 α( α) 2 τ m +(α α2 ) κ m +α2 κ m +( α)2 α = κ m α κm +, α( α) 2 τ m + +(α α2 ) κ m + +α2 κ m + +( α)2 α( α) 2 τ m +(α α2 ) κ m +α2 κ m +( α)2 It ca be cocluded from Eqs (78), (79) ad (70) that Propositio 72 Uder iteratios of defiite proportioal divisio poit, a space discrete cetroaffie curve with costat cetroaffie curvatures ad torsios is stable Especially, a affiely regular polygo is stable By previous coclusios[2], we have 27