a.i.bobenko, d.matthes and yu.b. suris- discrete and smooth orthogonal systems: c∞-approximation

8/3/2019 A.I.Bobenko, D.Matthes and Yu.B. Suris- Discrete and smooth orthogonal systems: C-approximation

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arXiv:math/0

303333v1[math.D

G]26Mar2003

Discrete and smooth orthogonal systems:

C-approximation

A.I. Bobenko D. Matthes Yu.B. Suris

Institut fur Mathematik, Technische Universitat Berlin,Str. des 17. Juni 136, 10623 Berlin, Germany.

EMails: [email protected] , [email protected],

[email protected]

1 Introduction and main results

Triply orthogonal coordinate systems have attracted the attention of math-ematicians and physicists for almost two hundred years now. Particularexamples of them were already used by Leibniz and Euler to evaluate mul-tiple integrals in canonical coordinates, and later Lame and Jacobi carriedout calculations in analytical mechanics with the help of the famous ellipticcoordinates. The first general results on the geometry of triply orthogo-nal systems date back to the 19th century like the famous theorem ofDupin, saying that coordinate surfaces intersect along curvature lines. TheLame equations which analytically describe the triply orthogonal systemswere studied in detail by Bianchi [Bi2] and Darboux [Da].

Recently orthogonal systems came back into the focus of interest inmathematical physics as an example of an integrable system. Zakharov[Z] has shown how the Lame equations can be solved by the -methodand constructed a variety of explicit solutions with the help of the dress-ing method. Algebro-geometric solutions of the Lame equations were con-structed by Krichever [K]. The recent interest to the orthogonal coordinate

systems is in particular motivated by their applications to the theory of theassociativity equations [Du].

Partially supported by the SFB 288 Differential Geometry and Quantum Physicsand the DFG Research Center Mathematics for Key Technologies (FZT 86) in Berlin.

Supported by the SFB 288 Differential Geometry and Quantum Physics.

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Figure 1: An elementary hexahedron of a discrete orthogonal system

The question of proper discretization of the classical models in differentialgeometry became recently a subject of intensive study (see, in particular,[BP, DS2, KS]). Indeed one can suggest various models in discrete geometry

which have the same continuous limit and nevertheless have quite differentproperties. For a great variety of geometric problems described by integrableequations it was found that integrable discretizations (i.e. the discretizationspreserving the integrability of the underlying nonlinear system) preserve thecharacteristic geometric properties of the problem. Moreover, it turns outthat this discretization can be described in pure geometric terms.

Such a discretization of the triply-orthogonal coordinate systems was firstsuggested by one of the authors [Bo]. It is based on the classical Dupin the-orem. Since the circular nets are known [MPS, N] to correspond to thecurvature line parametrized surfaces, it is natural to define discrete triply or-

thogonal systems as maps fromZ3

(or a subset thereof) toR3

with all elemen-tary hexahedrons lying on spheres. The neighboring spheres intersect alongcircles which build the coordinate discrete curvature line nets (see Fig.3).Doliwa and Santini [CDS] made the next crucial step in the developmentof the theory. They considered discrete orthogonal systems as a reductionof discrete conjugated systems [DS1], generalized them to arbitrary dimen-sion and proved their geometric integrability based on the classical Migueltheorem [Be]. The latter claims that provided the seven (black) points lieon circles as shown in Fig.1, the three dashed circles intersect in a com-mon (white) point. This implies that a discrete triply-orthogonal system isuniquely determined by its three coordinate circular nets (as shown in Fig.3).

This is a well-posed initial value problem for triply orthogonal nets.Later on discrete orthogonal systems were treated by analytic methods of

the theory of solitons. Algebro-geometric solutions [AKV] as well as the -method [DMS] were discretized preserving all the symmetries of the smooth

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Figure 2: Initial data for a smoothtriply orthogonal system.

Figure 3: Initial data for a discretetriply orthogonal system.

case.There is a common belief that the smooth theory can be obtained as a

limit of the corresponding discrete one, which can be treated also in thiscase as a practical geometric numerical scheme for computation of smoothsurfaces. Moreover, this discrete master theory has an important advantage:surfaces and their transformations appear exactly in the same way as sublat-tices of multidimensional lattices. This description immediately providesthe statements about the permutability of the corresponding (Backlund-Darboux) transformations, an important property which is nontrivial to

see and sometimes even difficult to check in the smooth theory.Until recently there were no rigorous mathematical statements supporting

the observation about the classical limit of the discrete theories. The first stepin closing this gap was made in our paper [BMS] where general convergenceresults were proven for a geometric numerical scheme for a class of nonlinearhyperbolic equations. The first geometric example covered by this theorywas smooth and discrete surfaces with constant negative Gaussian curvaturedescribed by the sine-Gordon equation.

In the present paper we formulate initial value problems and prove the cor-responding convergence results for conjugate and orthogonal coordinate sys-tems, which are described by three-dimensional partial differential/differenceequations. One should note here that despite the known geometric dis-cretization, a convenient analytic description of discrete orthogonal systemsconverging to the classical description of the smooth case was missing. Inparticular this was the problem with discrete Lame equations (cf. [AKV]).

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The Clifford algebra description of smooth and discrete orthogonal systemssuggested in [BH] turned out to be optimal for our purposes. This descrip-tion provides us with the proper discrete analogs of the continuous quantitiesand their equations.

In particular we introduce the discrete Lame system which is the start-ing point for further analytical investigations. We write these equations inthe form of a discrete hyperbolic system and formulate the correspondingCauchy problem. As a by-product, we rewrite the classical Lame systemin the hyperbolic form and pose a Cauchy problem for classical orthogonalcoordinate systems, which seems to be new.

The central theorems of this paper are on the convergence of discreteorthogonal coordinate systems to a continuous one. In particular we provethe following statement.

Theorem 1 Assume three umbilic free immersed surfaces F1, F2 andF3 inR3 intersect pairwise along their curvature lines, X1 = F2 F3 etc. Then:

1. On a sufficiently small box B = [0, r]3 R3, there is an orthogonal co-ordinate systemx : B R3, which has F1, F2 andF3 locally as imagesof the coordinate planes, with each curve Xi being parametrized locallyover the i-th axis by arc-length. The orthogonal coordinate system x isuniquely determined by these properties.

2. Denote by B = B (Z)3 the grid of mesh size inside B. Onecan define a family {x : B R3} of discrete orthogonal systems,parametrized by > 0, such that the approximation error decays as

supB |x

() x()| < C.

The convergence is C: all partial difference quotients of x convergewith the same rate to the respective partial derivatives of x.

We also answer the question of how to construct the approximating fam-ily {x}: One solves the discrete Lame system with discrete Cauchy datadetermined by F1, F2 and F3. The solution x

is then calculated in a processthat consists ofO(3)-many evaluations of the expressions in the discreteLame equations. The scheme can be and has been easily implementedon a computer, it needs a precision of 2 for the calculations and is robust

against C-deviations of order in the initial data.Actually, the results presented here go further:

We do not restrict ourselves to the case of three dimensions, but allowarbitrary dimensions, possibly different for domain and target space.

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In particular, we prove the corresponding approximation results forcurvature line parametrized surfaces.

One can keep some of the directions of the orthogonal system discretewhile the others become continuous. This way, the classical Ribaucour

transformation is obtained very naturally. The approximation resultholds simultaneously for the system and its Ribaucour transforms.

The results about the permutability of the Ribaucour transformationsare elementary in the discrete setup. Our approximation theorem im-plies the corresponding claim for classical orthogonal systems (Theorem8). The first part of this theorem is due to Bianchi [Bi1], the secondpart was proved in [GT] using Bianchis analytic description of theRibaucour transformations.

In addition to orthogonal systems, we also treat, in a completely anal-

ogous manner, conjugate nets. We pose a Cauchy problem for themand show that any continuous conjugate net, along with its transfor-mations, can be C-approximated by a sequence of discrete conjugatenets. The latter are quadrilateral lattices introduced by Doliwa andSantini [DS1].

The paper is organized as follows. We start in Sect. 2 with a short pre-sentation of the concepts developed in our recent paper [BMS], stating themain result about an approximation of solutions of a hyperbolic system ofpartial differential equations by solutions of the corresponding partial dif-ference system. In Sect. 3 we apply this theorem to conjugate systems.

We formulate a Cauchy problem for smooth and discrete conjugate systems,define the Jonas transformation and prove the convergence. Sect. 4 is dedi-cated to the description of orthogonal systems in Mobius geometry. Sect. 5contains the approximation results for orthogonal systems, a part of whichwas summarized above.

2 The approximation theorem

We start with a study of hyperbolic systems of partial difference and partialdifferential equations. A general convergence theorem is formulated, stating

that solutions of difference equations approximate, under certain conditions,solutions of differential equations, provided the approximation of equationsthemselves takes place.

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2.1 Notations

For a vector = (1, . . . , M) of M non-negative numbers, we define thelattice

P = (1Z) . . . (MZ) R

M. (1)

We refer to i as the mesh size of the grid P in the i-th direction; a direction

with a vanishing mesh size i = 0 is understood as a continuous one, i.e. inthis case the corresponding factor iZ in (1) is replaced by R. For instance,the choice = (0, . . . , 0) yields P = RM. We use also the following notationfor n-dimensional sublattices ofP (n M):

P

i1,...,in= { = (1, . . . , M) P

| i = 0 if i / {i1, . . . , in}} . (2)

Considering convergence problems, we will mainly deal with the followingtwo situations:

1. i = for all i = 1, . . . , M (all directions become continuous in the limit 0);

2. i = for i = 1, . . . , m; i = 1 for i = m + 1, . . . , M (the last M mdirections are kept discrete in the limit).

Thus, we always have one small parameter 0 only. In our geometric prob-lems, the first situation corresponds to the approximation of conjugate andorthogonal systems themselves, while the second situation corresponds to theapproximation of such systems along with their transformations. The under-lying difference equations for discretizations of conjugate (resp. orthogonal)systems, and for their transformations will be the same, the only distinction

being in the way the continuous limit is performed. We will distinguish be-tween these two situations by setting M = m + m, where m = 0 and in thefirst situation, while m > 0 in the second one.

Our considerations are local, therefore the encountered functions are usu-ally defined on compact domains, like the cube

B(r) =

RM | 0 i r for i M

(3)

in the continuous context, and the following compact subsets ofP in thelattice context. In the situation 1 (m = 0):

B(r) = { P | 0 i r for i M} ; (4)

in the situation 2 (m > 0):

B(r) = { P | 0 i r for 1 i m, i {0, 1} for m < i M} .(5)

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We will often use the notation

B0(r) = { (Z)m | 0 i r for 1 i m} , (6)

so that in the latter case B(r) = B0(r) {0, 1}m.

For lattice functions u : P

X with values in some Banach space X, weuse the shift operators i and the partial difference operators i = (i 1)/idefined as

(iu)() = u( + iei), (iu)() =1

i(u( + iei) u()) ; (7)

here ei is the ith unit vector. For i = 0, define i = 1 and i = i. Multipleshifts and higher order partial differences are written with the help of multi-indices = (1, . . . , M) N

M:

= 11 MM ,

= 11 MM , || = 1 + . . . + M. (8)

We use the following C-norms for functions u : B(r) X:

u = sup

|u()|X

B(r ||); || ; (9)if m > 0, only the multi-indices with i = 0 (i > m) are taken intoaccount in the last formula. Any function defined on B(r) with > 0 hasfinite C-norms of arbitrary order ; on the other hand, if u : B(r) Xis a function of a continuous argument which does not belong to the classC(B(r)), then for its restriction to the lattice points, u = uB(r), the normu will, in general, diverge for 0.

Definition 1 We say that a family of maps {u : B(r) X}0


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Let X = X1 XK be a direct product ofK Banach spaces, equippedwith the norm

|u()|X = max1kK

|uk()|Xk. (10)

Consider a hyperbolic system of first-order partial difference ( > 0) or partial

differential ( = 0) equations for functions u : P X:

juk = fk,j(u), 1 k K, j E(k). (11)

Thus, for each k = 1, . . . , K , there is defined a subset of indices E(k) {1, . . . , M }, called the set of evolution directions for the component uk; thecomplementary subset S(k) = {1, . . . M }\E(k) is called the set ofstatic direc-tions for the component uk. The dependent variables uk() = uk(1, . . . , M)are thought of as attached to the cubes of dimension #(Sk) adjacent to thepoint .

The functions fk,j are defined on open domains Dk,j() X and are of

class C there. We suppose that the following conditions are satisfied.

(F) The domains Dk,j() for 1 j m (that is for all directions whichbecome continuous in the limit 0) exhaust X as 0, i.e. anycompact subset K X is contained in Dk,j() for all k, for all j E(k),

j m, and for all small enough. Similarly, there exists an open setD X such that the domains Dk,j() for m + 1 j M correspondingto the directions remaining discrete in the limit, exhaust D as 0.The functions fk,j locally converge to the respective f

0k,j in C

(X), resp.

in C(D), for any , with the rate of convergence O().

The existence of the subset D reflects the fact that transformations of conju-gate, resp. orthogonal, systems are not defined for all initial data. The set Dconsists of good initial values, where the transformations are well defined.

Goursat Problem 1 Given functions Uk onPS(k) (the subspace spanned by

the static directions of uk), and r > 0, find a function u : B(r) X

that solves the system (11) and satisfies the initial conditions uk = Uk on

PS(k) B

(r).

In order for this problem to admit a unique solution for > 0, thishas to be the case just for one elementary M-dimensional cube B(). The

corresponding condition is called consistency, and it turns out to be necessaryand sufficient also for the global (formal) solvability of the above Goursatproblem. The consistency condition can be expressed as

j(iuk) = i(juk), (12)

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for any choice ofi = j from the respective Ek, if one writes out all the involvedus according to the equations (11). The first step in writing this out reads:

j(fk,i(u)) = i(fk,j(u)), (13)

In order for the lefthand side (say) of this equation to be welldefined, thefunction fk,i is allowed to depend only on those u for which j E. This hasto hold for all j Ek, j = i, so we come to the condition that fk,i dependsonly on those u for which Ek \ {i} E. If this is satisfied,then the finalform of the consistency condition reads:

1j

fk,i(u + jfj(u)) fk,i(u)

= 1i

fk,j(u + ifi(u)) fk,j(u)

, (14)

where we use the abbreviation fi(u) for the Kvector whose kth componentis fk,i(u), if i Ek, and is not defined otherwise.

It is readily seen that, if one prescribes arbitrarily the values of all vari-ables uk at = 0 (recall, uk(0) is actually associated with the respectivestatic facet of dimension #(Sk) adjacent to the point = 0), then the system(11) defines uniquely the values for all variables on the remaining facets ofthe unit cube if and only if it is consistent. The following proposition showsthat the local consistency assures also the global solvability of the Goursatproblem; the only obstruction could appear if the solution would hit the setwhere the functions fk,j are not defined.

Proposition 1 Let > 0, and assume that the system (11) is consistent.Then either the Goursat problem 1 has a unique solution u on the whole ofB(r), or there is a subdomain B(r) such that the Goursat problem 1has a unique solution u on , and u() / D() at some point .

Finally, we turn to the whole family (0 0) of Goursat problems forequations (11) with the respective data Uk. The following result has beenproven in our recent paper [BMS].

Theorem 2 Let a family of Goursat problems 1 be given. Suppose that theyare consistent for all > 0, and satisfy the condition (F).

1. Let the functions U01 , . . . , U 0K be smooth, and, if m

> 0, assume thatthe point (U01 (0), . . . , U

0K(0)) belongs to the subsetD. Then there exists

r > 0 such that the Goursat problem 1 at = 0 has a unique solutionu : B(r) X.

2. If the Goursat data Uk locally O()-converge in C to the functions

U0k , then there exists 1 (0, 0) such that the Goursat problems 1 aresolvable on B(r) for all 0 < < 1, and the solutions u

: B(r) Xconverge to u : B(r) X with the rate O() in C(B(r)).

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3 Conjugate nets

We give definitions of continuous and discrete conjugate nets and their trans-formations, formulate corresponding Goursat problems, and prove conver-gence of discrete conjugate nets towards continuous ones.

3.1 Basic definitions

Recall that M = m + m. In Definition 2 below it is supposed that m = 0,M = m, while in Definition 3 it is supposed that m = 1, M = m + 1.

Definition 2 A map x : B0(r) RN is called an m-dimensional conjugate

net in RN, if ijx span(ix, jx) at any point B0(r) for all pairs1 i = j m, i.e. if there exist functions cij , cji : B0(r) R such that

ijx = cjiix + cijjx . (15)

Definition 3 A pair ofm-dimensional conjugate nets x, x+ : B0(r) RN is

called a Jonas pair, if three vectors ix, ix+ andMx = x

+ x are coplanarat any point B0(r) for all 1 i m, i.e. if there exist functionsciM, cMi : B0(r) R such that

ix+ ix = cMiix + ciM(x

+ x) . (16)

Remarks.

x+ is also called a Jonas transformation of x. One can iterate thesetransformations and obtain sequences (x(0), x(1), . . . , x(T)) of conjugate

nets x(t)

: B0(r) RN

. It is natural to think of such sequences as ofconjugate systems with m continuous and one discrete direction. Wewill see immediately that the notion of fully discrete conjugate systemsputs all directions on an equal footing.

A Combescure transformation x+ of x is a Jonas transformation forwhich vectors ix

+ and ix are parallel, i = 1, . . . , m. This class oftransformations is singled out by requiring ciM = 0 in equation (16).

Discrete conjugate nets were introduced by Doliwa and Santini [DS1].

Definition 4 A map x : B(r) RN is called an M-dimensional discrete

conjugate net inRN, if the four points x, ix, jx, and ijx are coplanarat any B(r) for all pairs 1 i = j M, i.e. if there exist functionscij : B

(r) R such that

ijx = cjiix + cijjx. (17)

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If m = 1, M = m + 1, then the M-dimensional discrete conjugate net,considered as a pair of functions x, x+ : B0(r) R

N, is also called a Jonaspair of m-dimensional discrete conjugate nets.

3.2 Hyperbolic equations for conjugate netsIntroducing M new functions wi : B

(r) RN, we can rewrite (17) as systemof first order:

ix = wi , (18)

iwj = cjiwi + cijwj , i = j, (19)

ickj = (jcik)ckj + (jcki)cij (ickj)cij , i = j = k = i. (20)

For a given discrete conjugate net x : B(r) RN, eq. (18) defines thefunctions wj , then eq. (19) reflects the property (17), and eq. (20) is just atranscription of the compatibility condition i(jwk) = j(iwk). Conversely,

for any solution of (18)(20) the map x : B(r) RN has the definingproperty (17) and thus is a discrete conjugate net. So, discrete conjugatenets are in a one-to-one correspondence to solutions of the system (18)-(20).

This system almost suits the framework of Sect. 2.2; the only obstructionis the implicit nature of the equations (20) (their righthand sides dependon the shifted variables like ickj which is not allowed in (11). We returnto this point later, and for a moment we handle the system (18)-(20) justas if it would belong to the class (11). In this context, we have to assign toevery variable a Banach space and static/evolution directions. Abbreviatingw = (w1, . . . , wM) and c = (cij)i=j , we set

X = RN{x} (RN)M{w} RM(M1){c},

with the norm from (10). We assign to x no static directions, to wi the onlystatic direction i, and to cij two static directions i and j. The correspondingGoursat problem is now formulated as follows.

Goursat Problem 2 (for discrete conjugate nets). Given:

a point X RN,

M functions Wi : Pi R

N on the respective coordinate axes Pi, and

M(M 1) functions Cij : Pij R on the coordinate planes P

ij,

find a solution(x, w, c) : B(r) X to the equations (18)-(20) satisfyingthe initial conditions

x(0) = X, wiPi= Wi , c

ijPij= C

ij. (21)

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We discuss now the consistency of the discrete hyperbolic system (18)(20). First of all, note that for any triple of pairwise different indices ( i,j,k)the equations of this system involving these indices only form a closed sub-set. In this sense, the system (18)(20) consists of three-dimensional buildingblocks. One says also that it is essentially three-dimensional. Correspond-ingly, in the case M = 3 consistency is not an issue: all conditions to beverified are automatically taken into account by the very construction of theequations (18)(20).

In this case M = 3 the initial data for an elementary cube are: a pointx(0), three vectors w1(0), w2(0), w3(0) which are thought of as attachedto the edges incident to the point = 0 and parallel to the correspondentaxes, and six numbers cij(0), 1 i = j 3, which are thought of asattached to the plaquettes incident to the point = 0 and parallel to thecoordinate planes Pij . From these data, one constructs first with the helpof (18) the three points ix = x(0) + iwi(0), then one calculates iwj for

i = j with the help of (19), and further jix by means of (18). Noticethat the consistency jix = ijx is assured by the symmetry of the righthand side of (19) with respect to the indices i and j. Thus, the first twoequations (18), (19) allow us to determine the values of x in seven verticesof an elementary cube. The remaining 8-th one, 123x, is now determinedas follows. Calculate ickj from the (linearly implicit) equations (20), thencalculate jiwk from (19), and then 123x from (18). Again, the consistencyrequirement jiwk = ijwk is guaranteed, per construction, by eqs. (20).Geometrically, the need to solve the linearly implicit equations for ickj isinterpreted as follows. The point 123x is determined by the conditionsthat it lies in three planes ijk , 1 i 3, where ijk is defined as the

plane passing through three points (ix, jix, kix), or, equivalently, passingthrough ix and spanned by iwj, iwk. All three planes 123, 213 and312 belong to the threedimensional space through x(0) spanned by wi(0),1 i 3, and therefore generically they intersect at exactly one point. Theabove mentioned linearly implicit system encodes just finding the intersectionpoint of three planes.

Turning now to the case M > 3, we face a nontrivial consistency prob-lem. Indeed, the point ijkx can be constructed as the intersection pointof three planes ijk, ikj , ijk , where the plane ijk is definedas the plane passing through three points (ijx, ijkx, ijx), or, equiv-

alently, passing through ijx and spanned by ijwk, ijw. So, there arefour alternative ways to determine the point ijkx, depending on whichindex plays the role of i.

Proposition 2 The system (18)-(20) is consistent for M > 3.

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Proof. The geometric meaning of this statement is that the above mentionedfour ways to obtain ijkx lead to one and the same result. This can beproved by symbolic manipulations with the equations (18)-(20), but a geo-metric proof first given in [DS1] is much more transparent. In the construc-tion above, it is easy to understand that the plane ijk is the intersectionof two threedimensional subspaces ijk and jik in the fourdimensionalspace through x(0) spanned by wi(0), wj(0), wk(0), w(0), where the sub-space ijk is the one through the four points (ix, ijx, ikx, ix), or,equivalently, the one through ix spanned by iwj , iwk and iw. Now thepoint ijkx can be alternatively described as the unique intersection pointof the four threedimensional subspaces ijk, jik, kij and ijk ofone and the same fourdimensional space.

Now, we turn to the above mentioned feature of the system (18)(20),namely to its implicit nature. The geometric background of this feature is

the following. Though there is, in general, a unique way to construct the8-th vertex ijkx of an elementary hexahedron out of the the seven verticesx, ix and ijx, or, equivalently, out of x, wi, cij , it can happen that theresulting hexahedron is degenerate. This happens, e.g., if the point ijxlies in the plane kij through (kx, ikx, jkx). This is illustrated on Fig.4. While the example on the left of Fig. 4 is fine, the right one does not

Figure 4: A non-degenerate and a degenerate hexahedron.

have the combinatorics of a 3-cube (the top-right edge has degenerated toa point). As a result, some of the quantities ickj in equation (20) are notwelldefined; this is reflected by the fact that the equations (20) are implicitand have to be solved for ickj , which is possible not for all values of c butrather for those belonging to a certain open subset.

To investigate this point, rewrite (20) as

ickj = F(ijk)(c) + j(jcik)ckj + j(jcki)cij i(ickj)cij (22)

Introducing vectors c and F(c) with M(M1)(M2) components labelled

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by triples (ijk) of pairwise distinct numbers 1 i,j,k M,

c(ijk) = ickj

F(ijk)(c) = cikckj + ckicij ckjcij,

we present the above equations as

(1 Q(c))c = F(c), (23)

with a suitable matrix Q(c). We need to find conditions for 1 Q(c) to beinvertible.

3.2.1 One conjugate net

If m = 0, then all entries of Q(c) are either zero or of the form cij .Therefore, on any compact K X we have: Q(c) < C with some

C = C(K) > 0, hence 1 Q(c) is invertible for < 0 = 1/C, and the theinverse matrix (1 Q(c))1 is a smooth function on K and O()-convergentin C(K) to 1. So, the system (23) is solvable, and

ickj = F(ijk)(c) + O() = cikckj + ckicij ckjcij + O(),

where the constant in O() is uniform on any compact set.The limiting equations for (18)-(20) in this case are:

ix = wi (24)

iwj = cjiwi + cijwj (25)

ickj = cikckj + ckicij ckjcij. (26)

For any solution (x,w,c) : B0(r) X of this system the function x : B0(r) RN is a conjugate net, because equations (24) and (25) yield the defining

property (15). Conversely, if x : B0(r) RN is a conjugate net, then equa-

tion (24) defines the vectors wi, these fulfill (25) since this is the defining prop-erty of a conjugate net, and (26) results from equating i(jwk) = j(iwk).So, the following is demonstrated:

Lemma 1 If m = 0, then the Goursat problem 2 for discrete conjugatenets fulfills the condition (F). The limiting system for = 0, consisting of(24)-(26), describes continuous conjugate nets.

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3.2.2 Jonas pair of conjugate nets

In the case m = 1 the matrix 1Q(c) is no longer a small perturbation of theidentity, since M = 1. More precisely, the matrix Q

(c) is blockdiagonal, its

M3 diagonal blocks Q

{ijk}(c) of size 6 6 correspond to nonordered triples

of pairwise distinct indices {i,j,k}, with rows and columns labeled by thesix possible permutations of these indices. Blocks for which i,j,k = M are ofthe form considered before their entries are either zero or of the form cij .Blocks where, say, k = M, admit a decomposition Q{ijM}(c) = A(c) + B(c),where the matrices A and B do not depend on . It is not difficult to calculatethat det(1 A(c)) = (1 + cMi)(1 + cMj). So 1 Q

{ijM}(c) is invertible if

cMi, cMj = 1, provided is small enough. The inverse is smooth in aneighborhood any such point c, and O()-convergent to (1 A(c))1.

As formal limit of (18) and (19), we obtain (i = j):

ix = wi (1 i m), (27)

Mx = wM, (28)

iwj = cjiwi + cijwj (1 i m), (29)

Mwj = cjMwM + cMjwj. (30)

There are also four different limits of equation (20), depending on whichdirections i and j are kept discrete. We shall not write them down; theyare easily reconstructed as the compatibility conditions for (29), (30), likeM(iwk) = i(Mwk) etc. Comparing (27)(30) with the definition of theJonas transformation (16), we see that the following is demonstrated.

Lemma 2 If m = 1, then the Goursat problem 2 for discrete conjugate netsalso fulfills the condition (F), with the domain

D = {(x,w,c) X | cMi = 1 for 1 i m}.

The limiting system at = 0 describes Jonas pairs of continuous conjugatenets.

3.3 Approximation theorems for conjugate nets

We are now ready to formulate and prove the main results of this chapter.

Theorem 3 (approximation of a conjugate net). Let there be given:

m smooth curves Xi : Pi RN, i = 1, . . . , m, intersecting at a common

point X = X1(0) = = Xm(0);

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for each pair 1 i < j m, two smooth functions Cij, Cji : Pij R.

Then, for some r > 0:

1. There is a unique conjugate net x : B0(r) RN such that

xPi= Xi , cijPij= Cij . (31)

2. The family of discrete conjugate nets {x : B0(r) RN}0 0:

1. In addition to the conjugate netx : B0(r) RN defined in Theorem 3,

there is its unique Jonas transformation x+ : B0(r) RN such that

x+Pi= X+i (1 i m); (36)

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2. The family {x : B(r) RN}0


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2. Given three Jonas transformations

x(, 1, 0, 0), x(, 0, 1, 0), x(, 0, 0, 1) : B0(r) RN

of a given mdimensional conjugate net x(, 0, 0, 0) : B0(r) RN, as

well as three further conjugate nets

x(, 1, 1, 0), x(, 0, 1, 1), x(, 1, 0, 1) : B0(r) RN

such that x(, 1, 1, 0) is a Jonas transformation of both x(, 1, 0, 0) andx(, 0, 1, 0) etc., there exists generically a unique conjugate net

x(, 1, 1, 1) : B0(r) RN

which is a Jonas transformation of all three x(, 1, 1, 0), x(, 0, 1, 1) andx(, 1, 0, 1).

These results are proved in the exactly same manner as Theorem 4, if onetakes into account the multi-dimensional consistency of discrete conjugatenets (Proposition 2).

4 Orthogonal systems

We give the definitions of orthogonal systems and their transformations, for-mulate corresponding Goursat problems, and prove convergence of discreteorthogonal systems to continuous ones.

4.1 Basic definitions

Definition 5 A conjugate net x : B0(r) RN is called an m-dimensional

orthogonal system inRN (an orthogonal coordinate system if m = N), if

ix jx = 0. (41)

Definition 6 A pair of mdimensional orthogonal systems x, x+ : B0(r) RN is called a Ribaucour pair, if for any 1 i m the corresponding

coordinate lines of x and x+ envelope a oneparameter family of circles, i.e.if it is a Jonas pair of conjugate nets and

ix+|ix+|

+ix

|ix|

(x+ x) = 0. (42)

Remarks.

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x+ is also called a Ribaucour transformation of x; Ribaucour trans-formations can be iterated, and our results generalize to any finitesequence of such transformations (cf. the remarks after Definition 3).

Two-dimensional (m = 2) orthogonal systems are called C-surfaces.

C-surfaces in R3 are characterized as surfaces parametrized along cur-vature lines.

There are different definitions of Ribaucour transformations in the lit-erature (cf. [H]). We have chosen the one that is best suited for ourpresent purposes.

Definition 7 A map x : B(r) RN is called anM-dimensionaldiscrete or-thogonal system inRN, if the four points x, ix, jx andijx are concircular

for all B(r) and all 1 j = j M.If m = 1, M = m + 1, then the M-dimensional discrete orthogonal

system, considered as a pair of functions x, x+ : B0(r) RN, is called aRibaucour pair of m-dimensional discrete orthogonal systems.

Remark.

Also in the discrete case we shall use the term C-surfaces for two-dimensional (m = 2) orthogonal systems.

So, orthogonal systems form a subclass of conjugate nets subject to a certainadditional condition (orthogonality of coordinate lines, resp. circularity ofelementary quadrilaterals).

The classical description of continuous orthogonal systems is this (in thefollowing equations it is assumed that i = j = k = i):

ix = hivi , (43)

ivj = jivi , (44)

ihj = hiij , (45)

ikj = kiij , (46)

iij + jji = ivi jvj . (47)

Eq. (43) defines a system {vi}1im of m orthonormal vectors at each point

B0(r); the quantities hi = |ix| are the respective metric coefficients.Conjugacy of the net x and normalization of vi imply that eq. (44) holdswith some realvalued functions ij . Eq. (45) expresses the consistencycondition i(jx) = j(ix) for i = j. Analogously, eq. (46) expresses theconsistency condition i(jvk) = j(ivk) for i = j = k = i. The m equations

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(45) andm3

equations (46) are called the Darboux system; they constitute

a description of conjugate nets alternative to the system (24)(26) we usedin Sect. 3. The orthogonality constraint is expressed by

m2

additional

equations (47), derived from the identity ijvi, vj = 0. In the case m = Nthe scalar product on the right-hand side of (47) can be expressed in termsof rotation coefficients ij only:

iij + jji +k=i,j

kikj = 0. (48)

The Darboux system (45), (46) together with the eqs. (48) forms the Lamesystem.

The classical approach as presented is based on the Euclidean geometry.However, the invariance group of orthogonal systems is the Mobius group,which acts on the compactification RN {} SN, rather than on RN.

This is a motivation to consider orthogonal systems in the sphereSN

and tostudy their Mobius-invariant description. In particular, this enables one togive a frame description of orthogonal systems which can be generalized tothe discrete context in a straightforward manner. This turns out to be thekey to derivation of the discrete analogue of the Lame system.

4.2 Mobius geometry

We give a brief presentation of Mobius geometry, tailored for our currentneeds. A more profound introduction may be found in [H].

The Ndimensional Mobius geometry is associated with the unit sphere

SN RN+1. Fix two antipodal points on SN, p0 = (0, . . . , 0, 1) and p =p0. The standard stereographic projection from the point p is a con-formal bijection from SN = S

N \ {p} to RN that maps spheres (of any

dimension) in SN to spheres or affine subspaces in RN. Its inverse map

1(x) =2

1 + |x|2x +

1 |x|2

1 + |x|2p0

lifts an orthogonal system in RN, continuous or discrete, to an orthogonalsystem in SN.

The group M(N) of N-dimensional Mobius transformations consists of

those bijective maps : SN SN that map any sphere in SN to a sphere ofthe same dimension; it then follows that is also conformal.

Fact 1 The notion of orthogonal system on SN is invariant under Mobiustransformations.

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Mobius transformations can be embedded into a matrix group, namely thegroup of pseudo-orthogonal transformations on the Minkowski space RN+1,1,i.e. the (N + 2)-dimensional space spanned by {e1, . . . , eN+2} and equippedwith the Lorentz scalar product

N+2i=1

uiei,N+2j=1

vjej

=N+1k=1

ukvk uN+2vN+2.

To do this, a model is used where points ofSN are identified with lines on alight cone

LN+1 =

u RN+1,1 | u, u = 0

RN+1,1.

This identification is achieved via the projection map

: RN+1,1 \ (RN+1 {0}) RN+1,

(u1, . . . , uN+1, uN+2) (u1/uN+2 , . . . , uN+1/uN+2),

which sends lines on LN+1 to points ofSN. In particular, the lines through

e0 =1

2(eN+2 + eN+1) and e =

1

2(eN+2 eN+1)

are mapped to the points p0 and p, respectively.The group O+(N + 1, 1) of genuine Lorentz transformations consists of

pseudo-orthogonal linear maps L which preserve the direction of time:

L(u), L(v) = u, v u, v RN+1,1, L(eN+2), eN+2 < 0. (49)

For L O+(N + 1, 1), the projection induces a Mobius transformation ofSN

via (L) = L

1

. Indeed, L is linear and preserves the light conebecause of (49), so (L) maps SN to itself. Further, L maps linear planesto linear planes; under , linear planes in RN+1,1 project to affine planes inRN+1. Since any sphere in SN is uniquely represented as the intersection of

an affine plane in RN+1 with SN RN+1, we conclude that spheres of anydimension are mapped by (L) to spheres of the same dimension, which isthe defining property of Mobius transformations. It can be shown that thecorrespondence L (L) between L O+(N + 1, 1) and M(N) is indeedone-to-one.

Recall that we are interested in orthogonal systems in RN; their liftsto SN actually do not contain p. Therefore, we focus on those Mobius

transformations which preserve p, thus corresponding to Euclidean motionsand homotheties in RN = (SN ). For these, we can restrict our attention tothe section

K =

u RN+1,1 : u, u = 0, u, e = 1/2

= LN+1 (e0 + e).

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(We use here the notation u for the hyperplane orthogonal to u.) Thecanonical lift

: RN K,

x x + e0 + |x|2e,

factors the (inverse) stereographic projection, 1 = , and is an isometryin the following sense: for any four points x1, . . . , x4 R

N one has:

(x1) (x2), (x3) (x4) = (x1 x2) (x3 x4). (50)

We summarize these results in a diagram

K LN+1 RN+1,1

RN1

SN RN+1

Denote by O+(N + 1, 1) the subgroup of Lorentz transformations fixing thevector e (and consequently, preserving K). The same arguments as beforeallow one to identify this subgroup with the group E(N) of Euclidean motionsofRN.

Fact 2 Mobius transformations of SN are in a one-to-one correspondencewith the genuine Lorentz transformations ofRN+1,1. Euclidean motions ofRN are in a one-to-one correspondence with the Lorentz transformations that

fix e.

Our next technical device will be Clifford algebras which are very convenient

to describe Lorentz transformations in RN+1,1, and hence Mobius transfor-mations in SN. Recall that the Clifford algebra C(N+ 1, 1) is an algebra overR with generators e1, . . . , eN+2 R

N+1,1 subject to the relations

uv + vu = 2u, v1 = 2u, v u, v RN+1,1. (51)

Relation (51) implies that u2 = u, u, so any vector u RN+1,1 \ LN+1

has an inverse u1 = u/u, u. The multiplicative group generated bythe invertible vectors is called the Clifford group. It contains the subgroupP in(N+ 1, 1) which is a universal cover of the group of Mobius transforma-tions. We shall need the genuine Pin group:

H = P in+(N + 1, 1) = {u1 un | u2i = 1},

and its subgroup generated by vectors orthogonal to e:

H = {u1 un | u2i = 1, ui, e = 0} H.

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H and H are Lie groups with Lie algebras

h = spin(N + 1, 1) = span

eiej : i, j {0, 1, . . . , N , }, i = j

, (52)

h = spin(N + 1, 1) = spaneiej : i, j {1, . . . , N , }, i = j. (53)In fact, H and H are universal covers of the previously defined Lorentzsubgroups O+(N + 1, 1) and O+(N + 1, 1), respectively. The covers aredouble, since the the lift (L) of a Lorentz transformation L is defined up toa sign. To show this, consider the coadjoint action of H on v RN+1,1:

A(v) = 1v. (54)

Obviously, for a vector u with u2 = 1 one has:

Au(v) = u1vu = 2u, vu v. (55)

Thus, Au is, up to sign, the reflection in the (Minkowski) hyperplane u

orthogonal to u; these reflections generate the genuine Lorentz group. Theinduced Mobius transformation on SN (for which the minus sign is irrelevant)is the inversion ofSN in the hypersphere (u LN+1) SN; inversions inhyperspheres generate the Mobius group. In particular, if u is orthogonalto e, then Au fixes e, hence leaves K invariant and induces a Euclideanmotion on RN, namely the reflection in an affine hyperplane; such reflectionsgenerate the Euclidean group.

Fact 3 Mobius transformations ofSN are in a one-to-one correspondence to

elements of the group H/{1}; Euclidean transformations ofRN are in aone-to-one correspondence to elements ofH/{1}.

4.3 Orthogonal systems in Mobius geometry

We use elements H as frames when describing orthogonal systems inthe Mobius picture. The vectors A(ei) for i = 1, . . . , N form an orthogonalbasis in TK at the point A(e0) K. For an orthogonal system x

: B(r) K, continuous ( = 0) or discrete ( > 0), with (m > 0) or without (m = 0)transformations, consider its lift to the conic section K:

x = x : B(r) K. (56)

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4.3.1 Continuous orthogonal systems

Define vi : B0(r) TK for i = 1, . . . , m by

ix = hivi, hi = |ix|. (57)

The vectors vi are pairwise orthogonal, orthogonal to e, and can be writtenas

vi = vi + 2x, vie (58)

with the vector fields vi from (43). As an immediate consequence, thesevectors satisfy

ivj = ji vi , 1 i = j m , (59)

with the same rotation coefficients ji as in (44).

Definition 8 a) Given a point x K and vectors vi TxK, 1 i m,

we call an element H suited to (x; v1, . . . , vm) if

A(e0) = x, (60)

A(ek) = vk (1 k m), (61)

b) A frame : B0(r) H is called adapted to the orthogonal systemx, if it is suited to (x; v1, . . . , vm) at any point B0(r), and

iA(ek) = kivi (1 k N, 1 i m). (62)

with some scalar functions ki.

Proposition 3 For a given continuous orthogonal system x : B0(r) K,an adapted frame always exists. It is unique, if m = N. If m < N, thenan adapted frame is uniquely determined by its value at one point, say = 0.An adapted frame satisfies the following differential equations.

Static frame equations:

i = eisi (1 i m) (63)

with si = (1/2)ivi.

Moving frame equations

i = Siei (1 i m) (64)

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with

Si = A1ei

(si) =1

2

k=i

kiek hie, (65)

where the functions ij and hi constitute a solution to the Lame system

(in the following equations the indices 1 i, j m and 1 k N arepairwise distinct):

ihj = hiij , (66)

ikj = kiij , (67)

iij + jji = k=i,j

kikj . (68)

Conversely, if ij and hi are solutions to the equations (66)-(68), then themoving frame equations (64) are compatible, and any solution is a frame

adapted to an orthogonal system.Proof. For an orthogonal coordinate system (m = N) the adapted frame isuniquely determined at any point by the requirements (60) and (61), whileeqs. (62) follow from (61). Ifm < N, extend the m vectors vi(0), 1 i m,by N m vectors vk(0), m < k N, to a orthonormal basis of Tx(0)K.There exist unique vector fields vk : B0(r) TK, m < k N with theseprescribed values at = 0, normalized (v2k = 1), orthogonal to each other,to vi, 1 i m, to x and to e, and satisfying differential equations

ivk = kivi , 1 i m, m < k N (69)

with some scalar functions ki. Indeed, orthogonality conditions yield thatnecessarily ki = ivi, vk. Eqs. (69) with these expressions for ki form awell-defined linear system of first order partial differential equations for vk.The compatibility condition for this system, j(ivk) = j(ivk) (1 i =

j m), is easily verified using (59). The unique frame : B0(r) H withA(e0) = x, A(ek) = vk (1 k N) is adapted to x.

Next, we prove the statements about the moving frame equation. Theequations

iA(e0) = hiA(ei), iA(ek) = kiA(ei)

are equivalent to

[e0, (i)1] = hiei , [ek, (i)

1] = kiei .

Since, by (53), (i)1 is spanned by bivectors, we come to the represen-

tation of this element in the form (i)1 = Siei with Si as in (65). The

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compatibility conditions for the moving frame equations, i(j) = j(i),are equivalent to

jSiej + iSjei + Si(eiSjei) + Sj(ejSiej) = 0,

which, in turn, are equivalent to the system (66)-(68). Conversely, for asolution : B0(r) H of the moving frame equations, define x = A(e0)and vk = A(ek). Then ix = hivi, and orthogonality of vi yields that x isan orthogonal system.

We conclude by showing the static frame equation. If is adapted, then(65) in combination with the orthogonality relation of the vi yields:

si = Aei(Si) =1

2

k=i

kiAei(ek) hiAei(e)

= 1

2

k=i

kiA(ek) + hiA(e) = 1

2

k=i

kivk + hie =1

2ivi .

The last equality follows from ki = ivi, vk and the fact that the ecomponent of ivi is equal to 2hi (the latter follows from (58)).

4.3.2 Discrete orthogonal systems

Now consider a discrete orthogonal system x : B(r) K. Recall that thecases m = 0 and m > 0 are treated simultaneously, so that i = for1 i m, and i = 1 for m + 1 i m + m

= M. Appropriate discreteanalogues of the metric coefficients hi and vectors vi are

hi = |ix|, vi = h1

i ix . (70)These are unit vectors, i.e. v2i = 1, representing the reflections taking x toix in the Mobius picture, cf. Fig. 5:

ix = Avix = x + ihivi . (71)

It is important to note that vectors vi are not mutually orthogonal. Next,we derive equations that are discrete analogues of (59). Since four points x,ix, j x and ij x are coplanar, the vectors jix and ijx lie in the span ofix and j x. The lift is affine, therefore the same holds for vis:

ivj

= (nji

)1 (vj

+ ji

vi). (72)

Here ji are discrete rotation coefficients, with expected limit behaviour ji ijji as 0, and

n2ji = 1 + 2ji + 2jivi, vj (73)

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vi

ivj

vj

j vi

x ix

ijxjx

Figure 5: An elementary quadrilateral of a discrete orthogonal system

is the normalizing factor that is expected to converge to 1 as 0. Cir-

cularity implies that the angle between ix and j x and the angle betweenjix and ijx sum up to :

jvi, vj + vi, ivj = 0. (74)

Under the coplanarity condition (72) the latter condition is equivalent to

vj(j vi) + vi(ivj) = 0. (75)

Inserting (72) and (73) into (74), one finds that either vi, vj = 1, i.e., thecircle degenerates to a line (we exclude this from consideration), or

ij+

ji+ 2

vi,

vj

= 0.

(76)

From (73) and (76) there follows:

n2ij = n2ji = 1 ijji . (77)

So, the orthogonality constraint in the discrete case is expressed as the con-dition (76) on the rotation coefficients; recall that in the continuous case thiswas a condition (47), resp. (48), on the derivatives of the rotation coefficients.

The following definition mimics the static frame equations of the contin-uous case.

Definition 9 A frame : B

(r) H is called adapted to a discrete or-thogonal system x, if

A(e0) = x, (78)

i = eivi (1 i M). (79)

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Existence of such a frame is a consequence of the circularity condition. In-deed, the consistency ji = ij is written as

eiejvj(j vi) = ejeivi(ivj).

which is equivalent to (75). An adapted frame is uniquely defined by thechoice of (0) (and thus is not unique, even if M = N). Indeed, eq. (79)together with the definition (70) of vi imply that if eq. (78) holds at onepoint of B(r), then it holds everywhere.

Proposition 4 An adapted frame for a discrete orthogonal system x sat-isfies the moving frame equations:

i = iei (1 i M) (80)

with

i = A1ei(vi) = Niei +i

2k=i

kiek ihie, (81)

where

N2i = 1 2i4

k=i

2ki , (82)

and the functions ij and hi solve the followingdiscrete Lame system (in theequations below 1 i, j M, 1 k N, and i = j = k = i):

ihj = (njij)1 (jhj + ihiij), (83)

ikj = (njij)1 (jkj + ikiij), (84)

iij = (njij)1 (2Niij jij), (85)

ij + ji = Niijj + Njjii ij

2

k=i,j

kikj . (86)

where the abbreviation nij = (1 ijji)1/2 is used, and ij are suitable real

valued functions. Conversely, given a solution ij and hi of the equations(83)-(86), the system of moving frame equations (80) is consistent, and itssolution is an adapted frame of a discrete orthogonal system.

Proof. From the definition i = A1ei

(vi) it follows that i is a vector

without e0component, i.e. admits a decomposition of the form (81). Again,from the definition of i and the moving frame equation (79) we find:

Aei(j) = eiejvj1ejei , ij = ejeivi(ivj)vi

1eiej .

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Upon using the identity vi(vj +ji vi)vi = vj +ij vi , which follows easily from(76), we can now represent eq. (72) in the following equivalent form:

nji(ij) = Aei(j) ijAej(i). (87)

This equation is equivalent to eqs. (83)(85). Next, the consistency j(i) =i(j) of the moving frame equations (80) is equivalent to

(ij)Aej(i) + (ji)Aei(j) = 0. (88)

Inserting the expressions for ij and ji from (87) into the left side give:

Aei(j)Aej (i) + Aej (i)Aei(j) + ji(Aei(j))2 + ij(Aej (i))

2

= Nijii + Njijj (ij/2)k=i,j

kikj (ij + ji) = 0,

the last equality being nothing but eq. (86). Conversely, eqs. (83)(86)imply (87), as well as the compatibility of the frame equations (80). So for anarbitrary initial value (0) H, the frame equations for : B

(r) Hcan be solved uniquely. Set x = A(e0), vi = Aei(i). From (87), (88) forthe quantities i there follow (72), (74) for the quantities vi. Using the factthat i has no e0component, one shows easily that there holds also (71).Therefore, x is a discrete orthogonal system.

4.3.3 Ribaucour transformation of a continuous orthogonalsystem

To describe a Ribaucour pair x, x+ : B0(r) K of continuous orthogonalsystems, one combines the descriptions of two previous subsections. Recallthat in the present context M = m + 1.

We denote by h+i , v+i ,

+ij the corresponding objects for x

+, defined as in(57), (59). Denote by vM unit vectors such that

x+ = AvM(x), (89)

cf. eq. (71). The defining property (42) of Ribaucour transformations andthe normalization of vM imply:

v+i = AvM(vi), (90)

ivM =i2

(v+i + vi) = i(vi vM, vivM), (91)

with m auxiliary functions i : B0(r) R.

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Definition 10 A pair of frames , + : B0(r) H is called adapted tothe Ribaucour pair x, x+, if is adapted to x and

+ = eMvM. (92)

Proposition 5 Let, + be a pair of frames adapted to the Ribaucour pairx, x+. Then the frame + is adapted to x+, and the following moving frameequations hold for +:

i+ = S+i ei

+ (1 i m), (93)

+ = MeM, (94)

where

S+i =1

2

k=i

+kiek h+i e (1 i m), (95)

M = A1eM

(vM) = NMeM +1

2

k=M

kMek hMe, (96)

N2M = 1 1

4

k=M

2kM, (97)

the functions hi, ij andh+i ,

+ij solve eqs. (66)-(68), and the following system

is satisfied (for 1 i = j m, 1 k N, and i = k = M = i):

h+i = hi + hMi , (98)

+

ki = ki + kMi , (99)+Mi = Mi + 2NMi , (100)

ihM =1

2(hi + h

+i )iM , (101)

ikM =1

2(ki +

+ki)iM , (102)

iiM = 1

2

k=i,M

(ki + +ki)kM + NM(Mi

+Mi) , (103)

ij =1

2i(ij +

+ij ) . (104)

Conversely, given solutions hi, ij and h+i ,

+ij of the system above with suit-

able auxiliary functions i, the moving frame equations (93), (94) are com-patible, and the solution , + is a pair of frames adapted to a Ribaucourpair of orthogonal systems.

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Proof. First we show that + is an adapted frame for x+. We have for1 i m, 1 k N:

A+(e0) = AvMAeM(e0) = AvM(x) = x+ ,

A+(e

i) = A

vMAe

M(e

i) = A

vM(v

i) = v+

i,

iA+(ek) = i(AvMAeM(ek)) = i(AvMA(ek))

= i(A(ek) 2vM, A(ek)vM)

= (ki 2ivM, A(ek))(vi 2vM, vivM)

= (ki 2ivM, A(ek))v+i =

+kiv

+i ,

where the minus sign applies iff k = M (and if k = 0, the latter calculationgoes through almost literally, with replacing ki,

+ki by hi, h

+, respectively).Next, from M = A

1eM

(vM) there follows:

kM = 2M, ek = 2vM, AeM(ek) = 2vM, A(ek),

hM = 2M, e0 = 2vM, AeM(e0) = 2vM, x,

NM = M, eM = vM, AeM(eM) = vM, A(eM).

This proves eqs. (98)(100). Further, eqs. (101)(103) are readily derivedby calculating the i-th paartial derivative of the respective scalar product.Finally, eq. (104) comes from the consistency condition i(j vM) = j(ivM).

Conversely, given a solution to the equations (66)(68) and (98)(104),the moving frame equations are consistent, thus defining the frames , +.It follows from Proposition 3 that both x = A(e0) and x

+ = A+(e0) areorthogonal systems. Furthermore, eq. (94), yields the defining relations

(89)(91) of the Ribaucour pair, with vM = AeM(M).

5 Goursat problems and approximation fororthogonal systems

It would be tempting to derive the theory of the Lame system (43)-(47)from its discrete counterpart (83)-(86), treating the latter as a hyperbolicsystem of the type (11). However, it turns out that in dimensions M 3this is hard to carry out, since one needs to enlarge the set of dependent

variables and equations in a cumbersome manner. The way around is basedon the following fundamental lemma, which allows one to take care of twodimensional orthogonal systems (in the coordinate planes) only, and to usethen the results for conjugate systems.

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Lemma 3 a) If for a conjugate net x, continuous or discrete, its restrictionto each plane Pij, 1 i = j M, is a C-surface, then x is a (continuous ordiscrete) orthogonal system.

b) If in a Jonas pair (x, x+), continuous or discrete, the net x is anorthogonal system, and the corresponding coordinate curves xPi and x

+Pienvelope onedimensional families of circles, then (x, x+) is a Ribaucour pairof orthogonal systems.

Proof in the discrete case is based on the Miguel theorem, cf. Sect. 1, andcan be found in [CDS]. For the continuous case it is a by-product of theproof of Theorem 6 below.

So, suppose that M = 2. Then the moving frame equations (80) and thesystem (83)-(86) take the form (1 i, j 2, 1 k N, i = j = k = i):

i = Ni 1i

1

2k=i

kiekei + hieei , (105)

ihj =ijjn

hi +1 n

inhj , (106)

iij =2Niijijn

1 + n

inij , (107)

ikj =1 n

inkj +

ijjn

ki, (108)

12 + 21 = 2N112 + 1N221 12. (109)

Here we use the abbreviations

n = n12 = n21 =

1 1221, =1

2

k>2

k1k2, N2i = 1

2i4

k=i

2ki .

(110)We show how to re-formulate the above system in the hyperbolic form andto pose a Cauchy problem for it.

5.1 Hyperbolic equations and approximation forC-surfaces

First consider the case m = 2, m

= 0, 1 = 2 = , corresponding to C-surfaces. Set

12 = N112 (2/2)( ), (111)

21 = N221 (2/2)( + ), (112)

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with a suitable function : P12 R. It can be said that the auxiliaryfunction splits the constraint, making the Lame system hyperbolic. Thissplitting plays a crucial role for our approximation results. In the smoothlimit 2 = 112 221 (cf. eqs. (114), (115) below).

As a Banach space for the system take

X = H{} R2(N1){} R2{h1, h2} R{} ,

where H is the space of an (arbitrary) matrix representation ofSpin(N+1, 1)

(note that if H H at some point, then eq. (105) guarantees thatthis is the case everywhere), denotes the collection of kj with k = 1, 2,1 j N, k = j. Obviously,

N1 = 1 + O(2), N2 = 1 + O(

2), n = 1 + O(2), (113)

where the constants in Osymbols are uniform on compact subsets ofX.

Both directions i = 1, 2 are assumed to be evolution directions for ; forhi and ki there is one evolution direction j = i, while for the additionalfunction both directions i = 1, 2 are static.

Goursat Problem 3 (for discrete C-Surfaces). Given H, func-tions Hi : P

i R and B

ki : P

i R for i = 1, 2, 1 k N, k = i, and

: P12 R, find a solution to the equations (105)-(108) with (111), (112),and with the Goursat data

(0) = , hiPi= Hi , kiPi= B

ki, P12=

.

Lemma 4 The hyperbolic system (105)(108) with (111), (112) is consis-tent. Goursat problem 3 for discrete C-surfaces satisfies condition (F). Thelimiting system for = 0 consists of equations (64) with (65), (66), (67) forpaiwise distinct indices 1 i, j 2, 1 k N, and

112 = (1/2)k>2

k1k2 + , (114)

221 = (1/2)k>2

k1k2 , (115)

which describe continuous C-surfaces.

Proof. Consistency is easy to see: the only condition to be checked is12 = 21, but this has already been shown in Proposition 4. Also otherstatements are obvious from (113). Notice that hyperbolic equations (114),(115) come to replace the nonhyperbolic orthogonality constraint (68).

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Now we discuss the way to prescribe the data in the Goursat problem3 in order to get an approximation of a given smooth Csurface. Unlike inthe case of general conjugate nets, it is not possible to prescribe the discretecurves Xi = x

Pi coinciding with their continuous counterparts Xi = xPiat the lattice points. However, it is still possible to achieve that X

i

arecompletely determined by Xi.

Given a curve Xi : Pi K, one has the corresponding tangential vectorfield iXi, and therefore the function hi = |iXi| and the field of unit vectorsvi = h

1i iXi. Take H suited to (Xi(0); vi(0)). Then by Proposition

3, there is a unique frame : Pi H adapted to the curve Xi with(0) = ; this frame is defined as the unique solution of the differentialequation i = Siei with Si given by eq. (65). According to the proof ofProposition 3, the latter equation is equivalent to the system of equations

ivk = kivi, ki = vk, ivi, k = i.

So, in order to determine the rotation coefficients ki : Pi R for 1 k N,k = i, one has to solve the latter system of ordinary differential equationswith the initial data vk(0) = A(ek). Thus, we produced the functions hiand ki, or, what is equivalent, the Clifford elements

Si =1

2

k=i

kiek hie

for a given curve Xi : Pi K. We say that hi, ki are read off the curve Xi.

Definition 11 The canonical discretization of the curve Xi : Pi K with

respect to the initial frame H suited to (Xi(0); vi(0)) is the functionXi = A(e0) : P

i K, where

: Pi H is the solution of the discretemoving frame equation i

= iei with

i =

1

2

4

k=i

2ki

1/2ei +

2

k=i

kiek hie

and the initial condition (0) = .

In other words, for the canonical discretization the data hi and ki are readoff the continuous curve. Obviously, is an adapted frame for the discrete

curve Xi . As 0, the canonical discretization Xi converges to Xi withthe rate O() in C.

Proposition 6 (approximation for Csurfaces). Letm = M = 2, andlet there be given:

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two smooth curves Xi : Pi RN (i = 1, 2), intersecting orthogonally

at X = X1(0) = X2(0),

a smooth function : P12 R.

Assume H is suited for (

X; v1(0), v2(0)). Then, for some r > 0:

1. There exists a unique Csurface x : B0(r) RN that coincides with

Xi onPi B0(r) and satisfies

112 221 = 2.

2. Consider the family of discrete Csurfaces {x : B0(r) RN}0


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m with Sij = Sji. Assume that for a given 1 i m all surfaces Sijintersect along the curvature lines Xi = Sij Pi. Set hi = |iXi|, assumethat hi(0) = 0, and set further vi = h

1i iXi. All curves Xi intersect at one

point X = X1(0) = . . . = Xm(0) orthogonally. Let H(N) be suitedto (X, v1(0), . . . , vm(0)). Construct the discrete Csurfaces S

ij

: Pij

RN

according to Proposition 6, with the functions ij = (iij jji)/2 for1 i < j m. Then for some r > 0:

1. There exists a unique orthogonal system x : B0(r) RN, coinciding

with Sij onPij B0(r).

2. There exists a unique family of discrete orthogonal systems {x : B0(r) RN}0


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5.3 Goursat problems and approximation forRibaucour transformations

Now we consider again a twodimensional (M = 2) discrete orthogonal sys-tem described by eqs. (105)(108), but perform a different continuous limit,

leading to two curves enveloping a circle congruence (Fig. 7). In other words,

Figure 7: A pair of curves enveloping a circle congruence

we set m = m = 1, 1 = , 2 = 1. Recall that in this case

B(r) = B0(r) {0, 1}, where B0(r) = P

1 [0, r] = (Z) [0, r],

so that only two lines in P12 are considered. Therefore, a function on P12

is conveniently considered as a pair of functions on P1, or even as just onefunction on P1, if one is interested in B

0(r) {0} only (this will be the case

for the function below). We denote the shift 2 by the superscript +.Set

21 = , 12 = N112 + (N221 ); (116)

with a suitable function : P1 R.As the Banach space where the system lives we choose, exactly as before,

X = H{} R2(N1){} R2{h1, h2} R{} .

In the present case we have:

N21 = 1 + O(2), N22 = 1

1

4

k=2

2k2 , n = 1

212 + O(

2) , (117)

where the constants in Osymbols are uniform on compact subsets of X.However, the system itself is now well defined not on all ofX but rather on

the subset where N22 > 0:

D =

( , ,h1, h2, ) :k=2

2k2 < 4

X. (118)

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Goursat Problem 4 (for a pair of curves enveloping a circle con-gruence). Given H, functions H

1 : P

1 R and B

k1 : P

1 R for

2 k N, and A : P1 R, as well as the real numbers H2 and B

k2 for

1 k N, k = 2, find a solution to the equations (105)-(108) with (116)on B(r) with the Goursat data

(0) = , h1P1= H1, k1P1= B

k1, P1= A

,

h2(0) = H2, k2(0) = B

k2.

Lemma 5 The hyperbolic system (105)(108) with ( 116) is consistent.Goursat problem 4 satisfies condition (F) on the subset D. The limitingsystem for = 0 consists of equations (64) with (65) for i = 1, (80) with(81) for i = 2, and

1h2 = 12(h1 + h2/2) , (119)

1k2 = 12(k1 + k2/2) , k > 2 , (120)

112 = 2(N221 ) + 212/2 , (121)

h+1 h1 = h2 , (122)

+k1 k1 = k2 , k > 2 , (123)

+21 21 = 2(N2 21) , (124)

and describes a pair of continuous curves enveloping a circle congruence.

Proof. Consistency is shown exactly as in Lemma 4, the limiting system iscalculated directly from (105)-(108) using (117). One sees that the systemof the lemma coincides with (98)-(103) for M = 2.

Proposition 7 (approximation of a pair of curves enveloping a circlecongruence). Letm = 1, M = 2, and let there be given:

a smooth curve X : P1 RN,

a smooth function A : P1 R,

a point X+(0) RN.

Set H2(0) = |X+(0) X(0)|, v2(0) = H2(0)

1(X+(0) X(0)). Assumethat H is suited for (X1(0); v1(0)), and set vk(0) = A(ek) for k = 2.Then, for some r > 0:

1. There exists a unique curve X+ : P1 B0(r) RN through the point

X+(0) such that the pair (X, X+) envelopes a circle congruence, and

|1X+| |1X| = A |X

+ X| . (125)

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2. Consider the family of pairs of discrete curves (X, (X)+)0


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1. Given an mdimensional orthogonal system x(, 0, 0) : B0(r) RN

and its two Ribaucour transformations x(, 1, 0) : B0(r) RN and

x(, 0, 1) : B0(r) RN, there exists a oneparameter family of orthog-

onal systems x(, 1, 1) : B0(r) RN that are Ribaucour transforma-

tions of both x(, 1, 0) and x(, 0, 1). Corresponding points of the fourconjugate nets are concircular.

2. Given three Ribaucour transformations

x(, 1, 0, 0), x(, 0, 1, 0), x(, 0, 0, 1) : B0(r) RN

of a given mdimensional orthogonal system x(, 0, 0, 0) : B0(r) RN,

as well as three further orthogonal systems

x(, 1, 1, 0), x(, 0, 1, 1), x(, 1, 0, 1) : B0(r) RN

such that x(, 1, 1, 0) is a Ribaucour transformation of both x(, 1, 0, 0)

and x(, 0, 1, 0) etc., there exists generically a unique orthogonal system

x(, 1, 1, 1) : B0(r) RN

which is a Ribaucour transformation of all three x(, 1, 1, 0), x(, 0, 1, 1)and x(, 1, 0, 1).

5.4 Example: elliptic coordinates

The simplest nontrivial example, to which the above theory can be applied, isthe approximation of two-dimensional conformal maps by circular patterns.

Starting with a conformal map F :R2

R2

, i.e. M = N = 2,h = |1F| = |2F| and 1F 2F = 0,

one calculates the metric and rotation coefficients according to equations (43)and (45), and the quantity according to (114):

h1 = h2 = h, 12 =1h

h, 21 =

2h

h, = 112 = 221

To construct a discrete approximation of F, solve the Goursat problem 3for > 0 with the data Hi , B

ij and

that are close to the values of therespective functions hi, ij and at the corresponding lattice sites. A good

choice is, for example, to prescribe

H1(1) = h(1 + /2, 0), H2(2) = h(0, 2 + /2),

B21(1) = 21(1 + /2, 0), B12(2) = 12(0, 2 + /2),

(1, 2) = (1 + /2, 2 + /2),

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Figure 8: Approximation of elliptic coordinates, = /10 and = /20.

for 1, 2 = i, 0 i R. If : B(R) H is the frame of the respectivesolution to the system (105)-(108), then the function

F : B(R) R2, F(1, 2) = ((1, 2)1e0(1, 2))

is a two-dimensional discrete orthogonal system, i.e., the points F(1, 2),F(1 + , 2), F

(1, 2 + ) and F(1 + , 2 + ) lie on a common circle

C(1, 2) in R2, and F(1, 2) = F(1, 2) + O().

This is illustrated with the planar elliptic coordinate system:

F(1, 2) = (cosh(1) cos(2), sinh(1) sin(2)),

whose coordinate lines are ellipses and hyperbolas. One findsh =

sinh2(1) + sin

2(2)1/2

,

12 = sinh(21)/(2h2), 21 = sin(22)/(2h

2),

= (1 cosh(21) cos(22)) /(4h4).

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Results are displayed in Fig. 8. Their left sides show the original coordinatelines of F, and on the right sides the circles C(1, 2) are drawn; each inter-section point of two coordinate lines on the left corresponds to an intersectionpoint of four circles on the right. There are small defects (the circles do notclose up) on the very right of the pictures since, in contrast to F, the discretemaps F are not periodic with respect to 2.

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[Bo] A.I. Bobenko, Discrete conformal maps and surfaces, In: Symmetriesand Integrability of Difference Equations, Proc. SIDE II Conference,Canterbury, July 1-5, 1996, Eds. P.A. Clarkson, F.W. Nijhoff, Cam-bridge Univ. Press, 1999, pp. 97108.

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