The increased degrees of freedom in a multitouch interface help users control 4D worlds with intuitive gestures. Relative to a specialized mouse-based interface, this multitouch interface is easier to learn, and 4D object manipulation is up to 1.5 times faster.

Navigation through 3D spaces is required in many interactive graphics and virtual reality applications. The authors consider the subclass of situations in which a 2D device such as a mouse controls smooth movements among viewpoints for a "through the screen" display of a 3D world. Frequently, there is a poor match between the goal of such a navigation activity, the control device, and the skills of the average user. They propose a unified mathematical framework for incorporating context-dependent constraints into the generalized viewpoint generation problem. These designer-supplied constraint modes provide a middle ground between the triviality of a single camera animation path and the confusing excess freedom of common unconstrained control paradigms. They illustrate the approach with a variety of examples, including terrain models, interior architectural spaces, and complex molecules.

This article reviews the available methods for automated identification of objects in digital images. The techniques are classified into groups according to the nature of the computational strategy used. Four classes are proposed:(1) the simplest strategies, which work on data appropriate for feature vector classification,(2) methods that match models to symbolic data structures for situations involving reliable data and complex models,(3) approaches that fit models to the photometry and are appropriate for noisy data and simple models, and (4) combinations of these strategies, which must be adopted in complex situations. Representative examples of various methods are summarized, and the classes of strategies with respect to their appropriateness for particular applications.

We present a new approach to the problem of modeling smoothly deformable shapes with convex polyhedral bounds. Our hyperquadric modeling primitives, which include superquadrics as a special case, can be viewed as hyperplanar slices of deformed hyperspheres. As the original hypersphere is deformed to its bounding hypercube, the slices undergo corresponding smooth deformations to convex polytopes. The possible shape classes include arbitrary convex polygons and polyhedra, as well as taperings and distortions that are not naturally included within the conventional superquadric framework. By generalizing Blinn's “blobby” approach to modeling complex objects, we construct single equations for nonconvex, composite shapes starting with our basic convex primitives. Hyperquadrics are of potential interest for the generation of synthetic images, for automated image interpretation and for psychological …

Advances in mathematics and physics have often occurred together. The development of Newton’s theory of mechanics and the simultaneous development of the techniques of calculus constitute a classic example of this phenomenon. However, as mathematics and physics have become increasingly specialized over the last several decades, a formidable language barrier has grown up between the two. It is thus remarkable that several recent developments in theoretical physics have made use of the ideas and results of modern mathematics and, in fact, have elicited the direct participation of a number of mathematicians. The time therefore seems ripe to attempt to break down the language barriers between physics and certain branches of mathematics and to re-establish interdisciplinary communication (see. for example, Robinson [1977]; Mayer [1977]).

In an attempt to find gravitational analogs of Yang-Mills pseudoparticles, we obtain two classes of self-dual solutions to the euclidean Einstein equations. These matries are free from singularities and approach a flat metric at infinity.

Recent work on Euclidean self-dual gravitational fields is reviewed. We discuss various solutions to the Einstein equations and treat asymptotically locally Euclidean self-dual metrics in detail. These latter solutions have vanishing classical action and nontrivial topological invariants, and so may play a role in quantum gravity resembling that of the Yang-Mills instantons.

Computer graphics has proven to be a very attractive tool for investigating low-dimensional algebraic manifolds and gaining intuition about their properties [9]. In principle, a computer image of any manifold described by algebraic equations can be produced by numerically solving the equations [2] to generate a fixed tessellation, or by using equivalent ray-tracing techniques [5]. However, for high-performance interactive manipulation of a manifold, it is much simpler and more practical to have a parametric representation instead of an implicit equation that must be solved numerically; a significant additional feature of many parametric representations is that they embody symmetry information that can be used to further enhance the visualization process. Numerical solutions typically do not naturally emphasize natural structures of manifolds, are poorly behaved near singularities and selfintersections, and are more difficult to explore using other visualization tools such as submanifold selection, coordinate transformations, and deformations.

F () R l~\V () RIJ]'his\v'ork is an outgro\vth of a serIes of lectures given by one of us Cf. 1\..) under the auspices of the i\. ccaden1ia Nazionale dei Lincei in 1\. 0111e in the spring of 1974. It is intended to help fill the need for a unified treat111cnt of Dirac's approach to the canonical Han1iltonian forn1ldation of singular I.. a-grangian systen1s.\\7c have attelnpted as far as possible to refer to the original literature on the subjcct, but there have undoubtedly been S0111C inad\'crtant 0111issions, for\vhich\ve apologize.\\ie\vish especially to thank Peter Goddard and Giorgio l) onzano for their essential participation in the for111ulation of the" string lTIodcl" gi\" en here, and Karel Kuchar for per111itting us to use parts of his unpublished lecture notes at Princeton in Chapter 7.C. T'. is grateful to J....~.\\lhceler for much encouragcn1ent, and to the National Science foundation for support under grant GP 30799X to Princeton lJnivcrsity\vhile i\..]. 1-1. thanks the Institute for i~ d\'anced Study, the) J ational Science~---oundation, and the US i\t0111ic Energy COn1111ission for their support of various phases of this project.\\7e are indebted to. i\caclen1ic Press, Inc, the publishers of...~ nnals of Physics C:\".\T.), for perlnission to usc various sections of I-1anson and l\. eggc (1974) and Regge and Teitelboin1 (1974) in this,," ork. T\\'o of us (...-\. J. H. and 1'. R.) arc grateful to the...~ ccaden1ia X azionalc dei L. incci for the congenial hospitality enjoyed\vhile this\,'ork,,'as being prepared.

The classical theory of the free relativistic spherical top is first developed from a Lagrangian viewpoint. Our method allows the invariant mass to be an arbitrary function of the intrinsic spin. A canonical formalism is established following the approach suggested by Dirac for constrained Hamiltonian systems. There is a second arbitrary function in the theory, in addition to the usual one due to reparametrization invariance. The usual Newton-Wigner variables are supplemented by the Euler angles. The quantum theory of the free top is discussed. The classical theory is generalized to included charged tops with magnetic moments.

A field theory is quantized covariantly on Lorentz-invariant surfaces. Dilatations replace time translations as dynamical equations of motion. This leads to an operator formulation for Euclidean quantum field theory. A covariant thermodynamics is developed, with which the Hagedorn spectrum can be obtained, given further hypotheses. The Virasoro algebra of the dual resonance model is derived in a wide class of 2-dimensional Euclidean field theories.

We examine the massless limit of a model for the massive relativistic Nambu string. The system possesses longitudinal kink modes excluded from the standard lightlike gauge treatment. We demonstrate the equivalence of these modes to those proposed by Patrascioiu. The classical nonlinear field theory of the two-dimensional string is shown to be a completely integrable Hamiltonian system. The Hamiltonian is expressed in terms of normal-mode action variables alone; the mass-squared spectrum is linear in the Bohr-Sommerfeld approximation. The difficulties of canonical quantization are exposed using a particular timelike gauge which admits commuting center-of-mass coordinates.

We develop a technique for attaching quark quantum numbers to world lines joined by relativistic strings. We are able to describe spin-0 and spin-½ U (n)-symmetric quarks attached to world lines. One spin-½ theory based on the Dirac equation yields a classical particle with helical motion, interpretable as Zitterbewegung. Another spin-½ model has no helical motion, but yields an algebra resembling that of super-symmetry. Motivated by duality diagrams and some general properties of quarkgluon models, we then construct quark-string models of mesons and baryons. The analysis of the meson model with unequal quark masses implies a stringlike spectrum with broken trajectory intercepts. A simple baryon model suggests a dynamical reason for diquark configurations in the lowest states. Physical weak and electromagnetic currents for the quark-string system follow from a minimal-coupling scheme as in gauge …

Visualization of uncertainty or error in astrophysical data is seldom available in simulations of astronomical phenomena, and yet almost all rendered attributes possess some degree of uncertainty due to observational error. Uncertainties associated with spatial location typically vary significantly with scale and thus introduce further complexity in the interpretation of a given visualization. This paper introduces effective techniques for visualizing uncertainty in large-scale virtual astrophysical environments. Building upon our previous transparently scalable visualization architecture, we develop tools that enhance the perception and comprehension of uncertainty across wide scale ranges. Our methods include a unified color-coding scheme for representing log-scale distances and percentage errors, an ellipsoid model to represent positional uncertainty, an ellipsoid envelope model to expose trajectory uncertainty, and a …