J. W. Bruce

Growth, motion and 1-parameter families of symmetry sets

Associated to every plane curve there is the locus of centres of circles bitangent to that curve, the so-called symmetry set of the curve. We can view this set as the spine of our curve, which can be recovered by taking the envelope of circles of varying radii along this spine. Varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object. Fixing the symmetry set and varying the radius function can be considered to model growth crudely. In this paper we shall describe the generic changes in the curves which take place in the process of growth and motion, and outline the corresponding results for spheres centred on a space curve. We also use the idea of a stratified Morse function to describe the generic changes which occur in one parameter families of (full) bifurcation sets in the plane. Applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set (and evolute) which occur in a generic isotopy of a plane curve.

Complete transversals and the classification of singularities

The classification of map-germs (up to a variety of equivalences) has many applications in differential geometry, to the study of wavefronts and caustics and to bifurcation theory. In a previous paper the first and last authors together with C T C Wall, gave some useful criteria for map-germs to be finitely determined with respect to a wide range of equivalence relations. The results presented in this paper, used in conjunction with these determinacy techniques, provide a very efficient classification procedure. Moreover, we show that the algebraic criteria involved in these calculations may be reduced to finite-dimensional symbolic problems which may be performed by a computer.

On binary differential equations

In this paper we give the local classification of solution curves of binary differential equations a(x,y)dy2+2b(x,y)dxdy+c(x,y)dx2=0 at points at which the discriminant function b2-ac has a Morse singularity. We also discuss the formal reduction of such equations to some normal form. The results determine the topological structure of asymptotic curves on a smooth surface with a flat umbilic, the principal curves at general umbilics, and asymptotic curves at cross-cap points of an otherwise smooth surface.

On binary differential equations and umbilics

In this paper we give the local classification of solution curves of bivalued direction fields determined by the equationwhere a and b are smooth functions which we suppose vanish at 0 ∈ ℝ2. Such fields arise on surfaces in Euclidean space, near umbilics, as the principal direction fields, and also in applications of singularity theory to the structure of flow fields and monochromatic-electromagnetic radiation. We give a classification up to homeomorphism (there are three types) but the methods furnish much additional information concerning the fields, via a crucial blowing-up construction.

Ridges, crests and sub-parabolic lines of evolving surfaces

The ridge lines on a surface can be defined either via contact of the surface with spheres, or via extrema of principal curvatures along lines of curvature. Certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications. There is a related concept of sub-parabolic line on a surface, also defined via extrema of principal curvatures.In this paper we study in detail the structure of the ridge lines, crest lines and sub-parabolic lines on a generic surface, and on a surface which is evolving in a generic (one-parameter) family. The mathematical details of this study are in Bruce et al. (1994c).

Parabolic curves of evolving surfaces

In this article we show how certain geometric structures which are also associated with a smooth surface evolve as the shape of the surface changes in a 1-parameter family. We concentrate on the parabolic set and its image under the Gauss map, but the same techniques also classify the changes in the dual of the surface. All these have significance for computer vision, for example through their connection with specularities and apparent contours. With the aid of our complete classification, which includes all the phenomena associated with multi-contact tangent planes as well as those associated with parabolic sets, we re-examine examples given by J. Koenderink in his book (1990) under the title of Morphological Scripts.We also explain some of the connections between parabolic sets and ‘ridges’ of a surface, where principal curvatures achieve turning values along lines of curvature.The point of view taken is the analysis of the contact between surfaces and their tangent planes. A systematic investigation of this yields the results using singularity theory. The mathematical details are suppressed here and appear in Bruce et al. (1993).

On families of square matrices.

In this paper we classify families of square matrices up to the following natural equivalence. Thinking of these families as germs of smooth mappings from a manifold to the space of square matrices, we allow arbitrary smooth changes of co-ordinates in the source and pre- and post- multiply our family of matrices by (generally distinct) families of invertible matrices, all dependent on the same variables. We obtain a list of all the corresponding simple mappings (that is, those that do not involve adjacent moduli). This is a non-linear generalisation of the classical notion of linear systems of matrices. We also make a start on an understanding of the associated geometry. 2000 Mathematics Subject Classification 58K40, 58K50, 58K60, 32S25.

Functions on a crosscap

The study of the differential geometry of surfaces in 3-space has a long and celebrated history. Over the last 20 years a new approach using techniques from singularity theory has yielded some interesting results (see, for example [3, 5, 19] for surveys). Of course surfaces arise in a number of ways: they are often defined explicitly as the image of a mapping f: R2[rightward arrow]R3. Since the subject is differential geometry one normally asks that these defining mappings are smooth, that is infinitely differentiable, however it is not true, in any sense, that most such parametrisations will yield manifolds. For such mappings have self-intersections, and more significantly they may have crosscaps (also known as Whitney umbrellas). Moreover if we perturb the maps these singularities will persist; that is they are stable. (See [14, 20] for details.) Consequently when studying the differential geometry of surfaces in 3-space there are good reasons for studying surfaces with crosscaps. In this paper we carry out a classification of mappings from 3-space to lines, up to changes of co-ordinates in the source preserving a crosscap. We can apply our results to the geometry of generic crosscap points. In [10] we computed geometric normal forms for the crosscap and used them to study the dual of the crosscap. We shall see that the approach here yields more information than that obtained in [10], although the latter has the advantage of being more explicit (in particular various aspects of the geometry can be compared using the normal forms). We refer the reader to [16, 18] for background material concerning singularity theory.

Dupin indicatrices and families of curve congruences

We study a number of natural families of binary differential equations (BDEs) on a smooth surface M in R-3. One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and principal BDEs, another between the characteristic and principal BDEs. The locus of singular points of the members of these families determine curves on the surface. In these two cases they are the tangency points of the discriminant sets ( given by a fixed ratio of principle curvatures) with the characteristic (resp. asymptotic) BDE. More generally, we consider a natural class of BDEs on such a surface M, and show how the pencil of BDEs joining certain pairs are related to a third BDE of the given class, the so-called polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDEs are intimately related.

Bifurcations of implicit differential equations

In this paper we give a local classi¯cation of the integral curves of implicit di®erential equations F (x; y; p) = 0; where F is a smooth function and p = dy=dx, at points where Fp = 0, Fpp =6 0 and where the discriminant f(x; y) : F = Fp = 0g has a Morse singularity. We also produce models for generic bifurcations of such equations and apply the results to the di®erential geometry of smooth surfaces. This completes the local classi¯cation of generic one-parameter families of binary di®erential equations (BDEs)