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Gordon Kindlmann, nowadays at the University of Chicago, seems to be the champion of tensor visualization by glyphs in the form of superquadrics (Kindlmann, 2004b). In his article at the Joint EUROGRAPHICS – IEEE Symposium on Visualization from 2004 (Kindlmann, 2004a), as well as in the presentation at the symposium, he concludes with a list of three future chores on which the specific work should concentrate. The first on this list is a “principled choice of γ”. I think that it is essential, actually the only task we need to have in view when dealing with tensor visualization by superquadric glyphs. Yet, enough amount of time went by, and no sign of “principled choice” in the specialty literature. Then again, it is perhaps only the fact that I personally attach so much importance to this issue. One can live with that, as one can live with anything that is unnecessary to life. But the general idea is that such a principled choice is in fact necessary to our very lives. And inasmuch as a principle should be involved in the choice, I dare to present one here.

First of all: what is the issue of “choice” we are talking about here? Because, if it turns out to be essential then perhaps this very fact is enough in order to urge the human spirit to a specific application. The principle would then do little, if anything, to the cause. On the other hand though, perhaps the importance of a principle on this issue is not quite so well perceived in the scientific community at large, because the work so far creates the impression that only very special groups of people need visualization by superquadric glyphs. In this case it doesn’t hurt much to make a little noise around, in order to draw attention to both the wide field of application of superquadric tensor glyphs, and their principle of choice, both mathematical and physical.

The visualization of anisotropic properties of matter is particularly easy in case these properties are represented by tensors, for the tensors are physically characterized by directions of most intense property – the eigendirections. In case the tensor is given by a symmetric 3×3 matrix it is even better: the eigendirections are orthogonal, and can thus picture an orthonormal frame of reference in every point of space where the tensor is defined. So, each point in space is the location of such a reference frame. If we don’t insist upon the symmetry of our tensor representation, or if the local property is manifested by a matrix in general, with no tensorial properties, we can even be contented with a non-orthogonal reference frame, as given by the eigendirections of the matrix. It all depends on the general physical properties of the location in space. However, the classical use of tensor properties teaches us that what we measure locally, at least in cases where the matter is involved, may well be a kind of average itself, due to the properties of isotropy of physical phenomena (Novozhilov, 1952). In order to illustrate this point let’s elaborate on the example that generated the idea first, and then speculate a little on the particle and light phenomena.

The statement that we can measure a certain multidimensional physical magnitude in a certain point of space envisions a highly idealized situation. First of all we are not always able to simultaneously measure multiple physical quantities, in view of the fact that these may interact in such a way that their measurements are mutually exclusive. A well known example is the one of conjugate variables in quantum mechanics. Nevertheless, we shouldn’t go that far with the imagination, for the most obvious example is provided by the image we make of the deformation of a continuum. Indeed, while from experimental point of view we can afford adequate pieces of matter to represent ideal states of deformation as close as possible to the standards we desire (uniaxial, biaxial, etc), inside a continuum the situation changes drastically. One cannot state, for instance, that in a certain point of that continuum there is a precise state of deformation of a kind or another. The most we can think of is a mixture of such states, and even that is a highly idealized situation, for we don’t know how the states of deformation coexist with each other: the description of a state of deformation or stress is a figment of our imagination. But, in the cases where these figments of our imagination are 3×3 matrices, the reason can always be conducted along the lines that follow, indicated by Novozhilov. A modern variant of these calculation is given by Kindlmann in his dissertation cited above (Kindlmann, 2004b, §5.6).

The idea is that a 3×3 matrix quantity defined in a point in space cannot be measured but by its intensities along directions and in planes through that point. This is a point of view which was instituted by Cauchy (for a review of the early specific works see Saint-Venant, 1872). The values of these intensities obviously vary with the direction and the plan of measurement. However, in a continuum, one can assume that, at least in certain ideal conditions of isotropy, the local manifestation of a matrix quantity is a certain average over all of the possible directions and planes through a point. When the matrix is a symmetric tensor, as one currently assumes in the theory of deformations, and furthermore, when one admits a uniform distribution of all directions and planes through a point in space, the averages over directions and planes can be given quite easily.

If x is our matrix, having the eigenvalues x_1,2,3, then its intensity along a certain direction given by the unit vector , can be calculated with the formula

(1)

where n^1,2,3 are the components of the unit vector of the direction in the system of eigendirections of x. So we can figure out that, in each one of the space points, a continuum can be characterized by an average of this quantity over the unit sphere. Representing the components of in terms of spherical angles as usual: sinθcosφ, sinθsinφ, cosθ, one can assume therefore that the continuum exhibits in any point the measurable mean

Performing this operation in (1), gives the well known value

(2)

On the other hand, if is the normal to a plane through a point inside a continuum, we can calculate the intensity of x on this plane, according to the formula representing Pythagoras’ theorem

(3)

Using the same procedure of averaging, we can find the average of this quantity in a point of the continuum as:

(4)

It is therefore to be expected that, in a continuum without inhomogeneities, when it comes to the measurement of a tensor, we only have at our disposal the quantities (2) and (4), in any of its points. And from these two quantities we ought to construct the eigenvalues of the tensor. Obviously then, the tensor is not uniquely defined. Even if the eigenvalues would be at our disposal, we still would have at least the arbitrariness of space rotations in the definition of a tensor. However the eigenvalues are not at our disposal, and we ought to construct them first, using just the quantities (2) and (4), and only after that bring in issues of space rotations.

In order to do this, we use a phase freedom, revealed in the construction of the roots of a cubic when one knows its Hessian. For, in case we measure the quantities (2) and (4) we have indeed at our disposal the root of the Hessian of the cubic equation whose roots are our eigenvalues x₁, x₂, x₃, i.e. the secular equation of the matrix x. This root is complex because the eigenvalues are supposed to be real, and is given by

(5)

One can see that the imaginary part of h is simply proportional with the magnitude of the shear vector in the octahedral plane. Therefore the orientation of this shear vector in the octahedral plane is arbitrary, and this is our gauge freedom. And even this freedom is eightfold ‘degenerated’ in a point in space, for there are eight planes in an octahedron. With these results, the formulas giving the eigenvalues x₁, x₂, x₃, define a one-parameter family of cubics, corresponding to the rotations of the shear vector in its octahedral plane. The situation is illustrated in the attached figure, where only the main octahedral plane is visualized.

The vectors here are having the three eigenvalues of the matrix as components, and the two thick vectors from our picture represent these eigenvalues in two different instances corresponding to the same measurement. Each one of these vectors can be decomposed in two components (the thin arrows), one in the octahedral plane, the other normal to the octahedral plane. The gauge freedom is then given by the angle Φ between the components from the octahedral plane. One can see that the figure may be constructed in any one of the eight octahedral planes, so the theory based on it is independent of the sign of eigenvalues. Each point in space can therefore be endowed with such an octahedron, and this seems to be quite a realistic representation of our physical world. The theory of plasticity may be cited here as witnessing the fact, but we think of a more meaningful example: the “eightfold way” of the modern theoretical physics (for the earliest works on this issue see Gell-Mann, Ne’eman, 1964).

It stands indeed the reason: the physical particles of our microscopic world have only short lifetimes, more precisely their existence cannot be but inferred in specific experiments over very limited portions of time. On the other hand, the mathematics allowed a systematic ‘census’ of these ‘elementary accidents’, done by means of algebraic structures, with the critical conclusion that if there are some fundamental bricks here, they should be confined inside matter. Any speculations on the confinement and deconfinement problems – of the quarks and partons, for instance – are always conducted along some lines, reminding (almost explicitly we should say!) of the true experimental origin of the problem, aptly baptized by Murray Gell-Mann as the eightfold way. In our terms described above, the explanation of this situation is just as simple as the following.

Ever since Fresnel started constructing the wave surface from bits and pieces accessible to experiment, it was quite clear that the technology has to enter necessarily the stage of human possibilities to describe the nature. The modern experimental particle physics is but the highest reflection of this philosophy. Indeed, we have no other possibility to reveal the properties of matter but by mediation of matter: there is no other way! In particle experiments the matter revealing their properties has specific forms: counters, cloud chambers, photo plates, and any modern, technologically enhanced, variation thereof. A particle penetrating matter in a specific experiment is visualized by a trajectory exposed by changing the properties of the revealing matter all along the motion of particle. One can figure out that the external entering particle perturbs the local state of stress in the revealing matter which, as shown above, is manifested by an octahedron like the one whose face is depicted in the figure above, in any point from the internal space of this matter. The perturbation can be described by a “phase lag” Φ in one of the eight planes of the local octahedron, whose ‘choice’ depends, of course, on the nature of the perturbing particle. Somehow, this purely experimental fact gets through into the census of particles in the form of an “eightfold way”. It is only natural then, that the idea of quark, generated by this way of visualizing the matter, should come along with the property of confinement, for the confinement is the essential property of the revealing matter in the first place. From this point of view, the forces of confinement should be described by the mathematical properties of stresses, rather than by Newtonian forces characterized by a potential.

But the most astounding example of the “eightfold way” is offered by the light itself, the one which stirred up the idea in the first place. It was realized quite early in the development of the theory of light, by James Mac Cullagh (Mac Cullagh, 1831), that the description of light should be done by what we would term today as a gauge theory, involving two vectors, rather than by the classical view, adopted by Fresnel himself, and involving the space arena. It is important, in hindsight, to notice that Mac Cullagh was forced to this assumption by the idea of reconciling the wave theory of light with the Newton’s theory of forces: the ellipse in the case of light couldn’t be explained as a Kepler orbit within Fresnel initial view. Now, inasmuch as the ether is involved here, Mac Cullagh’s is a tensor gauge, requiring a tensor which reminds us of the Maxwell stress tensor. The vacuum properties of light are then obtained by referring the scale angle Φ defined above to an explicit “quark arrangement” representing, this time, the natural deformation of space itself (Coll, Llosa, Soler, 2002), which seems to correlate the metric tensor directly to Newtonian forces.

However, the most beautiful part of this speculation still hangs in there for us: it is not necessary to use the symmetry in the tensor visualization of the world properties; it is not necessary to even use tensors! It is indeed sufficient to have matrices with real eigenvalues, for the theory of confinement applies directly in this case, with no other limitations apart from those related to Fresnel’s initial view, and amounting to the modern harmonic mappings. According to this view, we can associate with every complex number a Cauchy probability density. This complex parameter adopts a consistent estimate from three measurements of the Cauchy variable. This makes out of the complex parameter of the Cauchy distribution a root of the Hessian of the cubic having the three measured values as its own roots. Consequently, we can safely return to the subject matter that generated this discussion, with the safe conclusion of the general importance of a theory of visualization, whereby the three measured values can be anything physical, even the components of a vector. After all it was this observation that generated the idea of an “eightfold way” here in the first place. Only, instead of considering the eigenvalues as vectors, it is necessary to consider vectors as eigenvalues, that’s all. With a little care, the theory works, and one might even say that it is actually long standing (Chapman, 1891).

Now, after this short incursion into general reasons, hopefully the message that the whole fundamental physics needs visualization has been properly conveyed and we can concentrate on the main issue: the special visualization by superquadric tensor glyphs. Not being just as good with the visualization per se, it is better to leave this to Gordon Kindlemann’s work and exquisite presentations, and concentrate upon the mathematics of the problem. The best example of visualization is the one where the word applies literally so to speak: the light. Even since the times of Fresnel, we learned that the light is an expression of measure of space. It is expressed as a quadratic function of the ratios of three lengths along three orthogonal directions in a point in space:

(6)

Here x, y and z are the displacements ‘measured’ by the light motion in three orthogonal direction in a point in space, while a, b and c are the displacements permitted by the elasticities of ether along the very same directions. This choice was made by Fresnel based on the time variation of light, as it appeared in local experiments of diffraction: if it was to be described by dynamical laws, the closest model of this motion would be the undamped harmonic oscillator.

One can say that Fresnel created a visualization of the elasticity properties of the space based on the properties of light that correspond to the linear elasticity. This is actually the first visualization ever in physics, and it is still used today. However, if one does not use the accident that a periodic function is a solution of the second order differential equation just yet, a local function of the three ratios can very well be something like

(7)

or even something like

(8)

with λ, as well as α, β, γ, real but otherwise arbitrary. This means that, if we take the elasticity corresponding to light face value, it is a confinement force of ‘something’, like the confinement force referring to quarks and partons in nuclear and particle theories. The confinement forces are, however, never linear forces, but rather nonlinear, and even this quite in a specific way. This may also mean that the Newtonian dynamics should hardly work in the case of light. However, we shall confine our present argument to the Newtonian dynamics, for it has some merit – to say the least.

The equations (7) and (8) represent today’s superquadrics. Gordon Kindlmann insists on visualization of tensor properties with these superquadrics, rather than with regular quadrics like that from equation (6). But this is not the whole point: others have also insisted upon this kind of visualization. Kindlmann, however, shows that there is a merit in considering λ from equation (7), for instance, or α, β, γ from equation (8) for that matter, as functions of a certain measure of asymmetry of the very matrix representing the local properties. This shows indeed, both qualitatively and quantitatively, how anisotropic is a certain property represented by a matrix. Kindlmann’s idea can be roughly illustrated by an obvious example: that of comparison of an axisymmetric ellipsoid with a finite cylinder having the same axis of symmetry. The cylinder can be obtained as a superquadric of the same semi-axes as the ellipsoid. However, it is a lot more suggestive of the anisotropy of the property represented by the ellipsoid, especially when we consider that such a suggestion does not depend on the point of view, i.e. it is an intrinsic property just like the eigenvalues themselves. One can therefore realize that a “principled choice” of the exponents in equations (7) and (8), which come down to the choice of a single parameter in the Kindlmann’s original work, is of major importance in the general visualization of the world properties.

References

Chapman, C. H. (1891): On the Matrix which Represents a Vector, American Journal of Mathematics, Vol. 13, pp. 363 – 380

Coll, B., Llosa, J., Soler, D. (2002): Three-Dimensional Metrics as Deformations of a Constant Curvature Metric, General Relativity and Gravitation, Vol. 34, pp. 269 – 282

Gell-Mann, M., Ne’eman, Y. (1964): The Eightfold Way, W. A. Benjamin, Inc., New York

Kindlmann, G. (2004a): Superquadric Tensor Glyphs, In Proceedings of the VisSym 2004: Joint EUROGRAPHICS – IEEE TCVG Symposium on Visualization, pp. 147 – 154

Kindlmann, G. (2004b): Visualization and Analysis of Diffusion Tensor Fields, PhD Dissertation, School of Computing, University of Utah

Novozhilov, V. V. (1952): On the Physical Meaning of the Invariants of Tensions Used in the Theory of Plasticity (О Физическом Смысле Инвариантов Напряжения Используемых в Теории Пластичности), Prikladnaya Matematika i Mekhanika, Vol. 16, pp. 617–619

Saint-Venant, B. de (1872): Sur les Diverses Manières de Présenter la Théorie des Ondes Lumineuses, Gauthier-Villars, Paris