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## Über dieses Buch

This volume derives from a workshop on differential geometry, calculus of vari­ ations, and computer graphics at the Mathematical Sciences Research Institute in Berkeley, May 23-25, 1988. The meeting was structured around principal lectures given by F. Almgren, M. Callahan, J. Ericksen, G. Francis, R. Gulliver, P. Hanra­ han, J. Kajiya, K. Polthier, J. Sethian, I. Sterling, E. L. Thomas, and T. Vogel. The divergent backgrounds of these and the many other participants, as reflected in their lectures at the meeting and in their papers presented here, testify to the unifying element of the workshop's central theme. Any such meeting is ultimately dependent for its success on the interest and motivation of its participants. In this respect the present gathering was especially fortunate. The depth and range of the new developments presented in the lectures and also in informal discussion point to scientific and technological frontiers be­ ing crossed with impressive speed. The present volume is offered as a permanent record for those who were present, and also with a view toward making the material available to a wider audience than were able to attend.

## Inhaltsverzeichnis

### Multi-functions Mod ν

Abstract
We extend the theory of current valued multi-functions to multi-functions mod ν.
Frederick J. Almgren

### Computer Graphics of Solutions of the Generalized Monge-Ampère Equation

Abstract
Using the method of Minkowski or Alexandrov one finds simple discretizations of elliptic Monge-Ampère equations, including the equation of graphs with prescribed positive Gaussian curvature. It is shown how these discrete problems can be solved numerically, and computer graphics of the piecewise linear, convex solutions are presented.
Alfred Baldes, Ortwin Wohlrab

### Computer Graphics Tools for Rendering Algebraic Surfaces and for Geometry of Order

Abstract
New developments in interactive computer graphics make it possible for mathematicians to approach old subjects in fresh ways. Occasionally the new approaches reveal additional insights into things already well understood from other viewpoints. This note give several examples of projects developed in collaboration with undergraduate students at Brown University in courses related to differential geometry. In each case, computer graphics techniques are used to investigate some geometric phenomenon, and in each case the difficulties encountered by the program reveal some feature of geometric interest.
Thomas F. Banchoff

### A Low Cost Animation System Applied to Ray Tracing in Liquid Crystals

Abstract
Animated movies of scientific graphics can be recorded on film with the low-cost system of hardware and software described here. The hardware consists of a 16mm camera, a stepper motor, and a simple cameramotor controller. The software is designed to produce bitmaps from graphical data, combine bitmaps into composite frames, and record frames onto film. The camera is fully controlled by the same graphics workstation that is used to display the images, so that fades and dissolves can be done by software with a camera not designed for such special effects. The graphical data generated on a supercomputer, is subsequently transferred to the workstation where it is stored and recorded frame-by-frame according to a configuration file. A variant of the software, which operates across local and wide area networks, makes use of network computing software to send computationally intensive tasks to a remote supercomputer or to other workstations in a distributed computing environment. We have used the system to simulate polarized light microscope images of liquid crystals according to a single-scattering ray-tracing theory.
J. R. Bellare, J. A. MacDonald, P. P. Bergstrom, L. E. Scriven, H. T. Davis

### From Sketches to Equations to Pictures: Minimal Surfaces and Computer Graphics

Abstract
In 1984, mathematician David Hoffman and computer scientist Jim Hoffman (no relation) sat before a computer screen and saw a crude rendition of a new minimal surface unfold itself. Inspired by the computer graphics, David Hoffman and William Meeks III were able to prove that this surface, which had been discovered by C. Costa, was embedded [5][7]. The surface then joined the classical examples of the plane, the catenoid and the helicoid as the only known examples of complete, embedded minimal surfaces of finite topological type.
Michael J. Callahan

### Nonuniqueness and Uniqueness of Capillary Surfaces

Abstract
It is shown that for any gravity field g and contact angle 7, an axially symmetric container can be found that differs arbitrarily little from a circular cylinder and can be half filled with liquid in a continuum of distinct ways, such that no two of the surface interfaces are mutually congruent and such that all of them are in equilibrium with the same mechanical energy. This answers affirmatively a question raised by Gulliver and Hildebrandt [1], who obtained such a container in the particular case g = 0, γ = π/2.
For a particular surface in the continuum, it is shown that the second variation of energy can be made negative by a non-axisymmetric perturbation under the volume constraint, and thus that the surface can be embedded in a one-parameter family of nonsymmetric surfaces bounding constant volume, with decreasing energy. As a consequence, a rotationally symmetric container deviating arbitrarily little from a circular cylinder is characterized, so that an energy minimizing configuration filling half the container exists but will not be symmetric.
Finally, conditions on the container curvature are given under which the symmetric configurations with prescribed volume are uniquely determined. In the special case for which the container is a sphere and g = 0, the symmetric configuration is unique and energy minimizing among all possible configurations.
Details of this-work are given in [2] and in [3].
Paul Concus, Robert Finn

### Static Theory of Nematic Liquid Crystals

Abstract
This will be an expository lecture on liquid eyrstal theory, covering some of the physical phenomena of interest, the formulation of related mathematical problems, and recent mathematical work concerning analysis of them, as well as numerical computations.
More commonly considered static problems involve minimizing energies of the form
$$\int_\Omega {\omega (n,\vartriangle n)dx}$$
with Ω a give domain in ℝ3, n a vector field, constrained to be of unit magnitude,
$$n \cdot n = 1$$
with Lagrangians of the form
$$2\omega(n,\vartriangle n) = {K_1}{(\nabla n)^2} + {K_2}{(n \cdot vur\ln )^2} + {K_3}{\left| {n \times cur\ln } \right|^2} + \alpha [tr\nabla {n^2} - {(\nabla n)^2}]$$
Here, the K’s and α are constants, restricted so that w is non-negative. The special case
$${K_1} = {K_2} = {K_3} = \alpha = K \Rightarrow 2\omega = K{\left| {\nabla n} \right|^2}$$
also arises in the theory of harmonic mappings. This has produced some fruitful exchanges of ideas, although various serious difficulties have been encountered, in trying to adapt analyses to more general forms of ω. Boundary conditions of physical interest do include those of the familiar Dirichlet type. Then, minimizers must exhibit singularities at points in Ω if the boundary values are smooth, but not of topological degree zero, and may do so if this degree is zero. Discussion of this topic will follow rather closely the survey by Cohen, et al., in reference [1].
Other mathematical topics to be discussed, and more, will be covered in summary articles by Kinderlehrer [2] and Lin [3]. For one thing, other boundary conditions of physical interest leave n free to take on any values consistent with the condition that it makes a fixed angle with the normal to ∂Ω. This tends to produce singularities at points on ∂Ω, possibly also in Ω; existence and regularity theory have been extended to include such problems. Also, theory sketched above is not very satisfactory for dealing with some observed line singularities or defects. The lecture will touch upon ideas used to modify theory to correct this, and some results which have been obtained, using it.
J. L. Ericksen

### The Etruscan Venus

Abstract
State of the art computers, such as the Silicon Graphics IRIS, have something to offer journeyman topologists that can otherwise be experienced only in the imagination. It is the wonder of animating, in real time, complicated deformations of topological surfaces, interactively! This paper reports a project at the National Center for Supercomputing Applications (NCSA) to visualize a regular homotopy of a Klein bottle immersed in 4-space. The shadow (projection) of this phenomenon in 3-space is a mapping homotopy between stable images of closed, one-sided (non-orientable) 2-manifolds called ovalesques. Such surfaces are generated by the prescribed motion of an oval (e.g. an ellipse) through space. Thus, the notion of an ovalesque is a projective generalization of a ruled surface. Recall that ruled surfaces are generated by straight lines.
George K. Francis

### Harmonic Mappings, Symmetry, and Computer Graphics

Abstract
A method for visually representing mappings from three dimensions into two is discussed and motivated by recent proofs that x/|x| has minimum energy. Harmonicity is characterized in terms of the geometry of the representation. The angle problem for harmonic mappings is introduced.
Robert Gulliver

### Video Based Scientific Visualization

Abstract
We describe techniques for producing video imagery for scientific visualization. These techniques involve variations of graphics algorithms, distributed computing, and a versatile, low cost, video movie making system. Although video can be used for single frame displays, its obvious advantage is for animated movies. Video movies are made by single frame animation from the output of modeling processes like time dependent, numerical simulations. Visualization algorithms are used to convert abstract data into a geometric form for graphical display. The system uses a distributed architecture and extensive data compression to permit the use of wide area, as well as local area networks connecting the systems generating the data, the graphics, and doing the video recording.
W. E. Johnston, D. W. Robertson, D. E. Hall, J. Huang, F. Renema, M. Rible, J. A. Sethian

### Bubbles, Conservation Laws, and Balanced Diagrams

Abstract
A bubble is one physical system which can be modelled by a constant mean curvature surface in Euclidean three space R 3. Guided by common experience with bubbles, a natural question has been:
Must a compact constant mean curvature surface M in R 3 be a round sphere?
Rob Kusner

### Liquid Menisci in Polyhedral Containers

Abstract
Knowledge of fluid surfaces having constant mean curvature and of their stability limits is a prerequisite for scientific research under microgravity conditions, i.e. in the Spacelab or in the forthcoming Space Station. For the stability of liquid menisci in edges and corners it is essential to know, whether they exhibit negative or positive mean curvature, i.e. whether the meniscus generates an underpressure or an overpressure. A corresponding classification of possible surfaces is attempted. The stability criterion for convex liquid surfaces in long solid edges is indicated and relevant results obtained in parabolic flights of a KC-135 aircraft are reported.
Dieter Langbein, Ulrich Hornung

### Can One Hear the Shape of a Fractal Drum? Partial Resolution of the Weyl-Berry Conjecture

Abstract
Several years ago, motivated in part by the challenging problem of studying the scattering of light from “fractal” surfaces, the physicist Michael V. Berry formulated a very intriguing conjecture about the vibrations of “drums with fractal boundary”. Extending to the “fractal” case a long standing conjecture of Hermann Weyl, he conjectured in particular that the high frequencies of such “fractal drums” were governed by the Hausdorff dimension of their boundary [1,2].
Michel L. Lapidus

### Steepest Descent as a Tool to Find Critical Points of ∫ k 2 Defined on Curves in the Plane with Arbitrary Types of Boundary Conditions

Abstract
We will explicitly compute the gradient of the total squared curvature functional on a space of parametrized curves of fixed or variable length satisfying arbitrary types of boundary conditions. We show how to turn the space of such curves into an infinite dimensional submanifold of an inner product space. The steepest descent will then be along the integral curves of the negative gradient vector field in this manifold. We will derive the gradients and the corresponding flow equations. In conclusion we use computer graphics to illustrate this process by following one such trajectory starting close to an unstable critical point and ending at a stable critical point.
Anders Linnér

### Geometric Data for Triply Periodic Minimal Surfaces in Spaces of Constant Curvature

Abstract
In this note we describe the use of geometric data for the construction of triply periodic minimal surfaces in R 3, S 3 and H 3. With a conjugate surface construction we obtain the Plateau solution of a fundamental piece for the symmetry group of the minimal surface. For some examples in R 3 a method of H. Karcher and M. Wohlgemuth has led to the Weierstraß formula.

### Embedded Triply-Periodic Minimal Surfaces and Related Soap Film Experiments

Abstract
Some aspects of the modelling of embedded triply-periodic minimal surfaces (ETPMS) by both soap films and plastic models are discussed. Eight new examples of possible ETPMS are introduced. Interference-diffraction patterns obtained by reflecting laser light from soap films with non-zero Gaussian curvature are described.
Alan H. Schoen

### The Collapse of a Dumbbell Moving Under Its Mean Curvature

Abstract
The collapse of a dumbbell moving under its mean curvature has attracted considerable attention. A new class of numerical algorithms has been developed recently that can follow hypersurfaces propagating with curvature-dependent speed in any number of space dimensions. The essential idea behind these algorithms is to view the propagating hypersurface as a particular level set of a higher dimensional function. The motion of this higher-dimensional function is described by a scalar Hamilton-Jacobi equation with parabolic right-hand-side. This equation may be easily solved using techniques borrowed from the solution to hyperbolic conservation laws. We demonstrate this technique applied to the problem of a dumbbell collapsing under its own mean curvature. Our results show the breaking of the handle, the developing singularity, and the collapse of the two separate sections.
J. A. Sethian

### Geometry Versus Imaging: Extended Abstract

Abstract
There are two quite distinct ways of making pictures with computers. The geometric way is quite widely understood—and often thought to be the only way. The imaging way is less intuitive—and leads to a different marketplace which is probably as large or larger than that for geometry. The terminologies, theories, and even heroes of the two worlds are quite distinct, and the hardware devices to implement them are strikingly different.
Alvy Ray Smith

### Constant Mean Curvature Tori

Abstract
Constant mean curvature tori in ℝ3 were first discovered, in 1984, by Wente [15]. These examples solved the long standing problem of Hopf [6]: Is a compact constant mean curvature surface in ℝ3 necessarily a round sphere? Hopf proved that if the surface is topologically a sphere then it must be round and Alexandrov [3] proved that if the surface is embedded then it must be a round sphere.
I. Sterling

### On Deformable Models

Abstract
Deformable models are a new class of physically-based modeling primitives for use in computer graphics, especially for animation. Deformable curves, surfaces, and solids are governed by the mechanical principles of continuous bodies whose shapes can change over time. These primitives are capable of a variety of behaviors, including elasticity, viscoelasticity, plasticity, fracture, conductive heat transfer, thermoplasticity, melting, and fluid-like behavior in the molten state. By numerically simulating the equations of motion that govern deformable models, while visualizing their state variables, we are able to create realistic animations depicting nonrigid objects in simulated physical worlds.
Demetri Terzopoulos

### Periodic Area Minimizing Surfaces in Microstructural Science

Abstract
An A/B block copolymer consists of two macromolecules bonded together. In forming an equilibrium structure, such a material may separate into distinct phases, creating domains of component A and component B A dominant factor in the determination of the domain morphology is area-minimization of the intermaterial surface, subject to fixed volume fraction. Surfaces that satisfy this mathematical condition are said to have constant mean curvature. The geometry of such surfaces strongly influences physical properties of the material, and they have been proposed as structural models in a variety of physical and biological systems. We have discovered domain structures in phase-separated diblock copolymers that closely approximate periodic constant mean curvature surfaces. Transmission electron microscopy and computer simulation are used to deduce the three-dimensional microstructure by comparison of tilt series with two-dimensional image projection simulations of 3-D mathematical models. Three structures are discussed: the first of which is the double diamond microdomain morphology associated to a newly discovered family of triply periodic constant mean curvature surfaces. Second, a doubly periodic boundary between lamellar microdomains, corresponding to a classically known surface (called Scherk’s First Surface), is described. Finally, we show a lamellar morphology in thin films that is apparently related to a new family of periodic surfaces.
Edwin L. Thomas, David M. Anderson, David C. Martin, James T. Hoffman, David Hoffman

### Types of Instability for the Trapped Drop Problem with Equal Contact Angles

Abstract
This paper continues the study of the stability of a liquid drop forming a bridge between two parallel planes in the absence of gravity (Figure 1) (see [2], [3], [4]). It is important to realize that the curves of contact between the free surface of the drop and the planar plates are assumed to be free. In other words, a perturbation of the drop which changes the wetted regions on the planes is admissible. The parameters which are determined physically are the volume of the drop and the angles of contact between the free surface and the two fixed planes.
Thomas I. Vogel
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