## Abstract

Given an exact solution of a partial differential equation in two dimensions, which satisfies suitable conditions on the boundary of the domain of interest, it is possible to deform the boundary curve so that the conditions remain fulfilled. The curves obtained in this manner can be patched together in various ways to generate a remarkably broad range of domains for which the boundary constraints remain satisfied by the initial solution. This process is referred to as boundary tracing and works for both linear and nonlinear problems. This article presents a general theoretical framework for implementing the technique for two-dimensional, second-order, partial differential equations with a general flux condition imposed around the boundary. A couple of simple examples are presented that serve to demonstrate the analytical tools in action. Applications of more intrinsic interest are discussed in the following paper.

## 1. Introduction

Boundary tracing is a conceptually straightforward method for generating new solutions of a partial differential equation system with boundary conditions from a known one. Suppose that we are given a particular solution of a system in a certain domain satisfying prescribed conditions on the boundary; in its simplest form tracing asks how the boundary might be deformed so that the solution continues to hold in the domain and the conditions remain satisfied. To illustrate the application of the idea consider:

The function *η*(*x*, *y*)=exp(−*y*) satisfies Helmholtz's equation ∇^{2}*η*=*η* in the half-plane *y*>0 subject to the constant flux boundary condition ∇*η*.** n**=1 on

*y*=0, where

**denotes the outward normal to the domain.**

*n**y*=0 is not the only curve on which the boundary constraint is satisfied. It is easily verified that ∇

*η*.

**=1 along the curve**

*n**y*=

*Y*(

*x*)=ln[cos(

*W*±

*x*)] with the upper sign taken for −

*W*<

*x*<0, the lower for 0<

*x*<

*W*, and

*W*is some constant 0<

*W*<

*π*/2, see figure 1

*a*. This deformation of the

*y*=0 boundary into the

*y*<0 region can be translated in

*x*and can be joined onto other indentations with different values of

*W*. In this way, we can construct a menagerie of indented boundaries with the property that the function

*η*(

*x*,

*y*)=exp(−

*y*) satisfies the prescribed boundary condition on all of them; figure 1 displays a small selection. It is interesting to note that the inverse problem of determining the location of the boundary given the physics described above and solution data in

*y*>0 is ill-posed, since the above solutions are all indistinguishable in

*y*>0. It is this non-uniqueness that tracing exploits to derive a broad range of domains.

Boundary tracing is the device we can use to construct all these new boundaries and details of the method in operation follow presently. This paper is our second on the topic. In the first, Anderson *et al*. (2006), we showed how the technique can be used to obtain new exact solutions of the Laplace–Young equation which governs the shape of the free surface of liquid in a container. That work was specific in the sense that its principal motivation was to show boundary tracing in operation for one particular equation. In contrast, here our objective is to consider the approach in a more general setting. Central to this is an appreciation of the underlying geometry as this is helpful for determining both the scope and limitations of the technique. For definiteness, in this work we describe boundary tracing for second-order partial differential equations; the results obtained can be generalized to higher dimensions and higher order systems of partial differential equations and a start on this has been made in the first authors' thesis, Anderson (2002). In order to show that boundary tracing is not merely a theoretical construct of little intrinsic interest, in the following paper (Anderson *et al*. 2007, hereafter referred to as II), we illustrate the application of the analytical tools developed here by reference to three particular equations: Helmholtz's equation, Poisson's equation and the constant mean curvature equation. Boundary tracing has much wider application than just these three problems but these are chosen to demonstrate some of the variety that boundary tracing can exhibit; many more examples are described in Anderson (2002).

It would be misleading to claim boundary tracing as an entirely new approach to tackling partial differential equation boundary value problems. It has been used successfully in a variety of contexts but in largely ad hoc numerical ways. For example, Anderssen *et al*. (1969) investigated the ion uptake of a growing single tap root by computing the shape of the root hair envelope as a function of time. McNabb & Wake (1991) used boundary tracing to estimate the time taken for a thermal system to transit between two steady states by conduction, while McNabb *et al*. (1991) determined the cooling times of pseudo-ellipsoidal objects with Newtonian surface cooling. We believe the current work represents the first attempt to put boundary tracing on a systematic footing and to present it as a technique of general usefulness.

The principal results of our work are described in §2 where the primary issue of determining the shape of domains that can be generated using boundary tracing procedures is considered. We have already mentioned that our description of boundary tracing is best developed using geometrical constructs and in §2 some of the key ideas are introduced with examples chosen to best illustrate these concepts. Section 2 is very much the ‘engine room’ of the paper and most of the theoretical ideas we shall need are contained there. In order to minimize the risk of giving the paper too dry a flavour, in §3 we give an overview of the practical application of the results obtained in preparation for the work described in II.

## 2. General theory

To account for a broad range of situations of interest, we shall develop the theory for the solutions of second-order partial differential equations within a general domain *Ω* possessing an outward normal ** n**. On the edge of

*Ω*we prescribe a boundary condition of the form(2.1)where

*F*is some given function. An attraction of this form of constraint is that many of the frequently encountered physically significant conditions, including Neumann, Dirichlet, Robin and surface contact conditions can all be expressed in this way. We re-emphasize at this point that boundary tracing seeks to determine those curves along which equation (2.1) is satisfied for given

*F*and prescribed exact solutions

*η*. In particular, the procedure makes no explicit reference to the underlying physics as contained in the field equation determining

*η*(

*x*,

*y*); all that is needed is a function and a boundary condition.

Given the solution *η*, the boundary condition (2.1) at any point P can be thought of as a dot product constraint for determining the possible orientations of curves, the *traced boundaries*, through P. This condition divides the domain into two regions: the *viable* tracing domain, where |∇*η*|≥|*F*|, and the *non-viable* tracing domain, in which |∇*η*|<|*F*|. Separating these two domains is a *terminal curve* along which(2.2)so that the normal to the traced curve boundary * n* and ∇

*η*is aligned around the terminal curve. It is also appropriate that terminal curves are so-called, for we shall find that traced curves begin or end on them.

To illustrate these concepts in terms of example 1.1 introduced earlier, where |∇*η*|=exp(−*y*) and |*F*|=1, we see that the terminal curve is *y*=0 and the viable domain is simply *y*≤0. The viable domain in this case lies outside the domain of the original problem determining *η*, which is frequently the case for tracing problems. A more complicated situation is provided by

The function *η*=sin *x*+sin *y* is a periodic solution of ∇^{2}*η*=−*η* satisfying ∇*η*.* n*=0 along the lines

*x*=(2

*m*+1)

*π*/2 and

*y*=(2

*n*+1)

*π*/2 for integer

*m*and

*n*.

*η*.

*=*

**n***c*. Then the terminal curve is given by (∇

*η*)

^{2}=

*c*

^{2}, so that for this particular

*η*we have cos

^{2}

*x*+cos

^{2}

*y*=

*c*

^{2}. The viable domain is then a function of

*c*, see figure 2. When

*c*=0 the entire plane is viable, but for 0<

*c*<1 although the viable domain remains connected it is now punctured by regularly spaced holes. Once

*c*=1 the plane splits into a checker-board of squares, alternating between viable and non-viable regions. When

*c*is increased further so that 1<

*c*<2, the viable domain consists of unconnected islands. As

*c*→2, the islands continue to shrink until they finally disappear at

*c*=2. For

*c*>2, there is no viable domain.

The boundary condition (2.1) determines the angle *α* between * n* and ∇

*η*as(2.3)so that there are two distinct directions for possible traced boundaries at any ordinary point inside the viable domain. These directions are symmetrically arranged on either side of ∇

*η*. Outside the viable domain no such directions exist, and as the terminal curve is approached from within the viable domain cos

*α*→1, so

*α*→0, and the traced curves approximate to the contours of

*η*. In addition to ordinary points, there may be singular points within the viable domain, associated with singularities of

*F*and/or

*η*.

The two families of traced curves within the viable domain can be obtained by integrating (2.3). Relative to a standard Cartesian coordinate system, the boundary condition (2.1) yields the ordinary differential equation(2.4)for the traced curves *y*=*y*(*x*), which can be integrated (if necessary numerically) away from an arbitrary starting point in the viable domain for any prescribed *η*(*x*, *y*). Exact solutions for *y*(*x*), or solutions expressed in terms of a simple integral, are available for a number of cases with simple symmetry. For instance, in the case of example 1.1, the tracing equation (2.4) integrates to give(2.5)for arbitrary *W*. These traced curves join smoothly onto the *x*-axis at |*x*|=*W*. Since any portion of the *x*-axis is also an acceptable boundary, we can generate domains by patching together curves of the type (2.5) and portions of the *x*-axis, leading to many possible domains as indicated earlier. In contrast, for example 2.1 (for which it is recalled that *η*=sin *x*+sin *y* and ∇*η*.* n*=

*c*), the tracing equation cannot be integrated exactly when

*c*≠0. Instead, some numerical solutions are displayed in figure 3, which shows how the traced curves populate the viable domain. The traced curves obviously cannot enter the non-viable regions (which we have seen are isolated patches when 0<

*c*<1, but the infinite plane with patches removed when 1<

*c*<2). For later purposes, note that in the 0<

*c*<1 case, there are two special families of straight line traced curves given by(2.6)where

*c*=cos

*d*determines

*d*as a function of

*c*. These lines touch the rounded corners of the terminal curves at the points (

*d*, (2

*n*+1)

*π*/2), ((2

*n*+1)

*π*/2,

*d*) as can be seen in figure 3

*b*.

Excepting ordinary singular points, the differential equation defining the traced curves is well behaved away from the terminal curve, but some care is required in deducing its properties as the terminal curve is approached. It is easiest to investigate this aspect by developing a geometrical description of the behaviour and then justifying our intuition with appropriate analysis. Near a terminal curve, as defined by equation (2.2), the tracing equation (2.4) is approximated by d*y*/d*x*=−*η*_{x}/*η*_{y}, confirming the result obtained earlier that both traced curves and the *η*-contour curve through a point P on the terminal curve share a common tangent. However, these traced curves will not normally have the same curvature at P and so will separate away from P.

Generally speaking the *η*=constant contour crosses the terminal curve at P, as in figure 4*a*. When this occurs, we shall deem P an *ordinary* terminal point. The two associated traced curves through P cannot enter the non-viable domain, and so must both end at P, creating a (ceratoidal) cusp. To analyse the traced curves near P, it is convenient to introduce the function(2.7)so that *Φ*(*x*, *y*)=0 identifies the terminal curve and *Φ*≥0 in the viable domain. Evidently close to an ordinary point on the terminal curve, the differential equation (2.4) can be written in the form(2.8)where the superscript 0 refers to values evaluated at P. Thus (as noted earlier), the two traced curves have the same slope at P as the *η*-contour through this point, but they differ in slope by order as one moves away from P. A detailed local analysis shows thatwhere , , *β*=*B*/*A*, and where for the purposes of this local calculation only, *x* measures the distance from the terminal curve into the viable domain.

Although the generic situation is that the *η*-contour at P passes through the terminal curve at P, this does not have to happen. It is possible for the contour to join, but not cross, the terminal curve at P, in which case P will be referred to as a *critical* terminal point. In this case, both the two traced curves and the *η*-contour itself, sharing as they do a common tangent, smoothly attach to the terminal curve at P. Critical terminal points might be isolated, or the entire terminal curve might consist of such points in which case we say we have a critical terminal curve.1 This situation, which often occurs in cases of strong symmetry, is displayed in figure 4 and appears in example 1.1. There the logarithmic traced solutions all smoothly attached to the terminal curve *y*=0. It will be seen later that boundary curves with cusps normally produce unacceptable boundaries so that critical terminal points and their associated boundary curves are particularly important as far as boundary tracing is concerned. The straight line traced curves, identified earlier for example 2.1 when 0<*c*<1, see equation (2.6), touch the terminal curves at critical terminal points.

### (a) Boundary curvature

General results concerning the curvature *κ* of traced curves are of importance, especially close to the terminal curve where the boundary curves and the *η*-contours share the same slope and so are indistinguishable to first order. Consider a traced boundary parameterized by arc length *s* and with tangent * t*. According to the Serret–Frenet formulae, we have(2.9)and it is a simple matter to show that for any differentiable function

*η*(

*x*,

*y*)(2.10)where

*H*is the Hessian of

*η*. To obtain the curvature of a traced boundary curve in terms of known quantities (by which we mean

*η*,

*F*,

*and*

**n***), we differentiate the boundary condition (2.1) with respect to arc length*

**t***s*to give(2.11)using equation (2.9), which can be written in the algebraically more convenient form(2.12)using equation (2.10); here

**n**^{T}denotes the transpose of

*. Within the viable domain ∇*

**n***η*.

*≠0, so that(2.13)determines the curvatures of each of the two traced curves through any ordinary point within this domain. Of course the two curves have different*

**t***and*

**n***and consequently generally have different curvatures.*

**t**As the terminal curve is approached, ∇*η*.* t*→0 so that

*κ*→∞ except in the degenerate case(2.14)in which case

*κ*remains undetermined by equation (2.12). The non-degenerate case occurs at all ordinary terminal points, where cusps occur as was anticipated from the earlier geometrical discussion.

The degenerate case occurs at critical terminal points. To see this, recall that at a critical terminal point P the two boundary curves, the *η*-contours and the terminal curve all locally coalesce, with the common normal given by * n*=∇

*η*/|∇

*η*|, see equation (2.2). This result also shows thatso that the algebraic condition (2.14) conveniently identifies critical terminal points. To obtain the boundary curvature at critical terminal points, we need to resolve the degeneracy in equation (2.12) by differentiating the boundary condition once more with respect to arc length to give(2.15)which can be expanded to give(2.16)where

*J*is the fully symmetric trilinear function given by(2.17)with

*H*

^{F}denoting the Hessian of

*F*. This seemingly complicated result is in fact easy to use. We have a quadratic in

*κ*(with coefficients dependent on third-order derivatives of

*η*) which determines the curvatures of the two boundary curves at critical terminal points. While convenient for calculation purposes, this result provides little geometric insight.

As a quick demonstration, we apply these results to example 1.1. The expression (2.13) gives *κ*=|cos(*W*±*x*)| for traced curves at ordinary points within the viable domain, a result that can be verified directly. On the terminal curve *y*=0, the curvature can be found using (2.16) which yields *κ*^{2}−*κ*=0, so that *κ*=0 or *κ*=1. The *κ*=0 case corresponds to the terminal curve being a valid boundary and the *κ*=1 result corresponds to the non-terminal traced curves. On the other hand, the findings for example 2.1 are algebraically complicated and not particularly enlightening. A better understanding of the geometry in this case is possible after we develop more technical machinery in §2*b*.

### (b) The boundary tracing manifold

An improved appreciation of the global topology and geometry of traced curves (and, as a by-product, much welcomed simplified calculations) follows if the traced curves are mapped onto the boundary tracing manifold defined by(2.18)Here, *z* is the obvious third dimension and it is remembered that *Φ* was introduced in equation (2.7). Real values for *z* exist only for points (*x*, *y*) in the physical plane lying within the viable tracing domain, so the manifold divides the *x*–*y* plane into its viable and non-viable parts and the intersection of the manifold with *z*=0 defines the terminal curve. The approximating tangent planes to the manifold around the terminal curve are vertical. Now if the traced curve corresponding to the positive square root in equation (2.4) is mapped onto the upper (positive *z*) branch of the manifold, and the negative square root curve onto the lower branch, then the manifold separates the two sets of tracing curves. Thus, a single traced curve on the manifold projects onto the two boundary curves in the *x*–*y* plane. The cusps that occur at all ordinary terminal points in the *x*–*y* plane then simply correspond to locally smooth curves on the manifold; ordinary terminal points are indeed ordinary on the manifold. The resulting field on the manifold is single-valued and differentiable everywhere except at singular points within the viable domain and at critical terminal points, see later. The global topology of the direction field can thus be simply determined by examining the behaviour near critical terminal points and singular points within the viable domain, and connecting up the curves. From a numerical point of view there are also considerable benefits to working on the manifold, thereby avoiding instabilities (as a result of the cusps) that arise close to terminal curves. The traced curves displayed in figure 3 were in fact obtained by setting up and solving the ordinary differential equations for the traced curves (*x*(*s*), *y*(*s*), *z*(*s*)) on the manifold, and then projecting onto the real plane.

Recall that at a critical terminal point P on the real *x*–*y* plane, the tangents to the terminal curve and the *η*=*η*(*P*) contour (and also the traced curves) coincide, so that the manifold and the *η*=*η*(*P*) (cylindrical) surfaces touch at P, see figure 5. The shape of the curves of intersection of these surfaces and thus the traced curves on the manifold near P will therefore be strongly dependent on the relative curvatures of these surfaces, as is evident from figure 5. A detailed analysis (appendix A) shows that traced curves on the manifold close to an isolated critical terminal point are approximated by curves on the tangent plane given explicitly by(2.19)where *X* is the (appropriately scaled) distance from P around the terminal curve, *κ*_{Φ} and *κ*_{η} are the curvatures of the terminal curve and the *η*-contour at the critical terminal point, ‖∇*Φ*_{0}‖=|∇*Φ*(0, 0)|, and the constant *c* identifies a specific curve. The terms (corresponding to terms in the defining differential equation for the traced curves (2.8)) are vanishingly small near the terminal curve but are significant, and separate out the two traced curve branches away from the terminal curve.

If *κ*_{Φ}<*κ*_{η} then, ignoring for the moment the terms, the traced curves on the manifold are ellipses which form closed loops around the critical point, which is isolated, see figure 5*a*. The corresponding traced curves in the actual *x*–*y* plane also surround the isolated critical terminal point in the viable domain. The two sets of traced curves on the real plane are indistinguishable to this order near P. The effect of the neglected terms in equation (2.19) is to introduce asymmetries between the curves on the upper and the lower manifold branches, resulting in spiral behaviour on the manifold and a separation of the projections on the real *x*–*y* plane with consequent cusps along the terminal curve in the neighbourhood of the isolated critical point. In this case, because the critical terminal point is isolated, no acceptable boundary curve can be traced through it.

On the other hand, if *κ*_{Φ}>*κ*_{η} then, again first ignoring the terms, the traced curves form a saddle point on the manifold, see figure 5*b*. There are two smooth curves (the separatrixes) that pass through the critical terminal point. The existence of the term introduces cusps in the real-plane projections except at the critical terminal point. The effect on the two smooth curves through the critical terminal point is to cause a modification of the traced curves on the manifold away from the critical point, but the local behaviour will not be significantly affected. The projections of the two branches onto the real plane will however separate so that in this case there will be two smooth curves through the critical point in the real plane, one or both of which may give rise to useful boundaries.

An important special (and degenerate) case arises when the terminal curve is also an *η*-contour so that *κ*_{Φ}=*κ*_{η} along the complete critical terminal curve. In this case, the saddle collapses and we recover the special symmetric case discussed separately. This situation occurs in example 1.1 where the manifold is given by *z*^{2}=exp(−2*y*)−1 with *y*≤0. Normally in such cases, as in example 1.1, it is a simple matter to determine the traced curve geometry.

For example 2.1 there are no singularities in the viable domain, so the location and nature of the critical terminal points determines the geometry of the traced curves. The traced curves on the manifold for 0<*c*<1 and 1<*c*<2 are shown in figure 6. For 1<*c*<2, the manifold consists of isolated balls which project down onto the isolated patches in the *x*–*y* plane seen earlier in figure 3*c*. The traced curves on the manifold wrap around like balls of string, displaying spiral-type terminal critical points at either end of the winding axis orientated at *π*/4 or 3*π*/4 depending on the location of the ball. The traced curves on the real plane thus have cusps along the terminal curve except at the critical terminal points which are isolated. When 0<*c*<1, the traced curves on the manifold begin and end on neighbouring rounded-off square terminal loops, see figure 6*a* (and also perhaps more clearly in figure 3*b*). To make sense of this, we examine the behaviour close to one of the periodic arrays of terminal curves; see figure 7 where the terminal curve together with nearby *η*-profiles are plotted. As noted earlier, critical terminal points are located where *η*-contours and the terminal curve are tangential. Evidently, there are critical terminal points at each of the corners and the midpoints of each of the terminal sides, and nowhere else; a result that can be confirmed algebraically using equation (2.14). In addition, it is evident that the curvature of the terminal curve *κ*_{Φ} is greater than that of the local *η*-contour *κ*_{η} at each of the corner points, and less at the midpoints, again easily confirmed algebraically. Traced curves on the manifold thus form saddles at the corners and spirals at the midpoints of the terminal curve. The conclusion is, there are no useful paths through the midpoints and two smooth traced curves passing through each of the corners, see figure 7. Using equation (2.16), it can be shown that the curvature of one of the traced curves is zero; these are precisely the straight line traced curves we identified earlier, see equation (2.6). Various combinations of the two smooth traced curves through the critical terminal point may be used to generate useful boundaries.

### (c) Identifying acceptable boundaries

Typically, a single traced curve by itself leads to either an uninteresting or an unphysical domain; for example, the positive branch solution (2.5) in example 1.1 does not by itself represent a possible domain boundary. However, by patching together parts of several traced boundaries one can construct a wide variety of interesting domains as we have already seen for example 1.1. It is important to be able to recognize which of the many possible combinations of traced curves lead to physically sensible domains and which do not, especially if the curves cannot be exactly determined. For example 1.1, recognizing sensible domains was not a great problem as the underlying geometry was simple, but rarely we are so fortunate, and example 2.1 is a case in point. The main difficulty is an interior/exterior domain recognition problem. To resolve such issues, we attach arrows to the boundary curves to indicate the location of the external domain relative to a particular traced curve on the real plane. We adopt the usual convention that the normal to the boundary points away from the interior of the domain, and the arrow on the boundary is such that the exterior domain is to the right, see figure 8. With this convention, the hallmark of a sensible domain made by patching together traced curves is that the direction of the various arrows should be continuous. To further aid visualization in the examples to follow, we will also denote the external zone by shading. Examples of realistic and non-realistic domains that can be generated from traced curves are shown in figure 8*a*.

Of major importance is the observation that for continuous functions *F*, no sensible domain can be constructed using both branches of the cusp formed at an ordinary terminal point, as can be seen in figure 8. The upshot is that normally such traced curves are unusable.2 In many cases, this may mean that no acceptable boundaries are available at all or that only traced curves through critical terminal points may be used to construct acceptable boundaries.

In example 2.1, and when 1<*c*<2, sensible boundaries cannot be constructed as all traced curves form cusps on the closed and isolated terminal curves. When 0<*c*<1, we have seen that the viable domain is connected and traced curves begin and end on the periodic array of terminal loops. Almost all of these end up in cusps along the edges of terminal loops which eliminates almost all combinations. However, there are straight line traced curves that touch and connect up the corner critical terminal points and form a grid out of which one may be able to construct sensible boundaries, but one needs to ensure that arrow continuity is maintained for any constructed domains. The arrows are included on figure 8*b* which demonstrates that while boundaries like the square ABCDA are not acceptable, the square abcda is physically reasonable. It is also evident that we can construct rectangular domains such as fgjkf. Furthermore, there is no need to restrict attention to connected domains; for example, the domain shown in figure 9*a* is sensible and indeed very interesting; it is far from obvious that we could find an exact solution for Helmholtz's equation with constant flux conditions in such an intricate multi-boundary domain. Evidently, although not completely unlimited, there is a great deal of scope in the choice of acceptable boundaries. Furthermore, the reader will recall that there are many other traced curves near the straight line boundaries (including the other critical point curves) which can be used in combination with the grid curves to construct sensible boundaries; figure 9*b* illustrates a modification that can be made to a simple square domain. It is remarkable that all these domains admit the one and the same *exact* solution of Helmholtz's equation subject to a unifying boundary condition. Given the interest in wave scattering around multiple obstructions with and without small domain modifications, the above results might find useful application.

## 3. Conclusion

In this paper we have sought to develop a theoretical framework for boundary tracing as it may be applied to second-order partial differential equations with general flux conditions. We have seen how geometric considerations lead to the introduction of a terminology useful for describing important aspects of tracing (including the notions of viable domains, terminal curves and ordinary and critical terminal points). The introduction of the boundary tracing manifold enables one to more easily determine the legitimacy of candidate traced curves. We have also seen how orientation issues need to be considered when constructing physically reasonable boundaries from traced curves. These various constructs and ideas have been illustrated by reference to two fairly simple examples. Often a naive application of the boundary tracing equation (2.4) leads to the required boundary determinations without need to delve into the general results: example 1.1 was of this type, as the topological structure on the manifold is simple. Sometimes, however, the analytical and topological tools developed in this paper are required to determine how traced curves negotiate singularities on the manifold and what combinations of traced curves generate acceptable boundaries. This was the case in example 2.1 with its periodic array of critical terminal points of saddle and spiral types.

It should be emphasized that, although the tracing curves may need to be numerically determined, the results obtained in this paper are analytic or topological and thus ‘exact’ in a theoretical sense. This exactness enables one to proceed with confidence with any associated numerical work. One can determine the presence and nature of any critical terminal points by simply checking for sign changes in the expressions **n**^{T}*H*** t**−∇

*F*.

*and*

**t***κ*

_{Φ}−

*κ*

_{η}around the terminal curve whose position and curvature are known to any precision. Using these results and corresponding ones for singularities in the viable domain one can determine the topography of the direction field on the manifold and, by projection, on the real plane. Viable boundaries can then be identified by checking for direction continuity. Secure in the knowledge that boundaries of a particular shape do exist, one can then proceed with computation.

It is possible to outline the general procedure that needs to be followed to apply boundary tracing in any particular case. Briefly, one needs to:

Identify the viable domain and the terminal curve using equation (2.2).

Determine the locations of any critical terminal points (using equation (2.14)) and any singular points within the viable domain.

Determine the natures of the critical and singular points on the manifold.

Find the boundary curve traces on the manifold and then, by projection, on the real plane.

Calculate the direction field and thus isolate possible boundaries.

Perhaps the most surprising feature of boundary tracing is the extraordinary range and shape of domains that can be generated by the judicious patching together of available traced curves. It should be emphasized that the patching together of curves does not require linearity, although should linearity be available then the range of possible domains associated with a particular partial differential equation with a prescribed boundary condition can be greatly increased by just considering different combinations of possible solutions. For example, an obvious question arising out of the work for example 2.1 is whether it is possible to construct new boundaries for the Helmholtz's equation with ∇*η*.* n*=

*c*when

*c*>2. There were no viable domains when we restricted our attention to the

*η*=sin

*x*+sin

*y*solution, but there are many other available solutions, like , and superposition can be used for such linear problems to greatly extend the possibilities. It is a matter of choosing the solution to fit the circumstance; examples are shown in II. One would guess, however, that the application of tracing to nonlinear problems is likely to be of especial importance given the lack of available analytical tools for such problems.

To conclude, we remark that although the theory we have discussed here is intriguing, its real potential value can only be ascertained by applying it to equations, domains and circumstances of physical interest. It is this aspect that will be addressed in II where we outline how boundary tracing can be used in three specific applications.

## Footnotes

↵† Present address: Rising Sun Research, 133 Gouger Street, Adelaide, South Australia 5000, Australia.

↵It is theoretically possible that only a portion of the terminal curve consists of critical points, but we have not found any examples of this.

↵Although a reasonable boundary might be constructed if

*F*has a jump discontinuity across the cusp.- Received January 23, 2007.
- Accepted May 4, 2007.

- © 2007 The Royal Society