Abstract This paper enhances the voxel-to-voxel radiance shooting, propagation, and scattering algorithm of Max [20]. It reduces the memory requirements by storing the radiance only on the occupied cells of a sparse voxel octree and by sampling only 24 propagation direction bins instead of 96 or more. It gives a modification of the propagation algorithm to compensate for the larger solid angle of the direction bins and allows for position-dependent fully anisotropic scattering. For a volume of $$n^3$$ n 3 voxels, the computation time is $$O(n^3(\mathrm {log}\,n)^{3/2})$$ O ( n 3 ( log n ) 3 / 2 ) .

The reflection of a bright light source on a dynamic surface such as water with waves can be difficult to render well in real time due to reflection aliasing and flickering. In this paper, we propose a solution to this problem by approximating the reflection direction distribution for the water surface as an elliptical Gaussian distribution. Then we analytically integrate the reflection contribution throughout the rendering interval time. Our method can render in real time an animation of the time integrated reflection of a spherical light source on highly dynamic waves with reduced aliasing and flickering.

One step in the reconstruction pipeline for creating scene geometry from multiple camera views is the fitting of 3-D planes to image segments. We optimize the four homogeneous coefficients of the equation of a plane, so that the homography it induces on a source image segment minimizes a cost function with photo consistency and edge consistency terms. The photo consistency term for a segment is defined as the summed squared pixel color differences with all target images in which it is visible. For the edges we find a set of sub pixel-precise Canny edge points from the source image. The edge consistency term considers all pairs of segments which are adjacent in 3D, and sums the squared 2D distances of the appropriate Canny edge points to the projection of the line in which the two segment planes intersect. We apply the L-BFGS non-linear optimization algorithm, which uses derivatives of the cost function with respect to the plane equation coefficients, and we derive analytic formulas for these derivatives for both terms in the cost function. An alternate optimization method uses the sparse Hessian matrix of second derivatives, and we compute these second derivatives as well. We test these plane optimizations on both synthetic and real image sequences.

Automatic fusion of aerial optical imagery and untextured LiDAR data has been of significant interest for generating photo-realistic 3D urban models in recent years. However, unsupervised, robust registration still remains a challenge. This paper presents a new registration method that does not require priori knowledge such as GPS/INS information. The proposed algorithm is based on feature correspondence between a LiDAR depth map and a depth map from an optical image. Each optical depth map is generated from edge-preserving dense correspondence between the image and another optical image, followed by ground plane estimation and alignment for depth consistency. Our two-pass RANSAC with Maximum Likelihood estimation incorporates 2D-2D and 2D-3D correspondences to yield robust camera pose estimation. Experiments with a LiDAR-optical imagery dataset show promising results, without using initial pose information.

ABSTRACT This paper presents a finite element method‐based framework for an object withering simulation modeled with heterogeneous material, such as fruits drying or decay. We introduce diffusion procedures for both the moisture content and decay spread, which are solved directly on a tetrahedral mesh representation of the fruit flesh. Then, we use the moisture content to control shrinkage through the initial strain, which is integrated into the Lagrangian dynamic equation, and solved with the finite element method. For the complex structure of the object, another fine triangle mesh is used to represent the skin, and its deformation is solved by a thin shell technique. To couple the motion between different layers of the fruit, a tracking force is used to pull the skin and drive its deformation together with the volume mesh. In comparison with the previous work, our method provides temporally and spatially varying parameters to model the complex phenomena of object withering. Moreover, the water diffusivity can also be given by user input to present various material properties of the cut section and skin‐covered area. Our algorithm is easy to implement and highly efficient in generating a realistic appearance for the withering effect. For a medium‐scale model, we can achieve interactive simulation. Copyright © 2012 John Wiley & Sons, Ltd.

Abstract Most 3D vector field visualization techniques suffer from the problem of visual clutter, and it remains a challenging task to effectively convey both directional and structural information of 3D vector fields. In this paper, we present a novel visualization framework that combines the advantages of clustering methods and illustrative rendering techniques to generate a concise and informative depiction of complex flow structures. Given a 3D vector field, we first generate a number of streamlines covering the important regions based on an entropy measurement. Then we decompose the streamlines into different groups based on a categorization of vector information, wherein the streamline pattern in each group is ensured to be coherent or nearly coherent. For each group, we select a set of representative streamlines and render them in an illustrative fashion to enhance depth cues and succinctly show local flow characteristics. The results demonstrate that our approach can generate a visualization that is relatively free of visual clutter while facilitating perception of salient information of complex vector fields.

Abstract Strange attractors of 3D vector field flows sometimes have a fractal geometric structure in one dimension, and smooth surface behavior in the other two. General flow visualization methods show the flow dynamics well, but not the fractal structure. Here we approximate the attractor by polygonal surfaces, which reveal the fractal geometry. We start with a polygonal approximation which neglects the fractal dimension, and then deform it by the flow to create multiple sheets of the fractal structure. We use adaptive subdivision, mesh decimation, and retiling methods to preserve the quality of the polygonal surface in the face of extreme stretching, bending, and creasing caused by the flow. A GPU implementation provides efficient visualization, which we also apply to other turbulent flows.

Isosurfaces are fundamental volumetric visualization tools and are generated by approximating contours of trilinearly interpolated scalar fields. While a complete set of cases has recently been published by Nielson, the formal proof that these cases are the only ones possible and that they are topologically correct is difficult to follow. We present a more straightforward proof of the correctness and completeness of these cases based on a variation of the Dividing Cubes algorithm. Since this proof is based on topological arguments and a divide-and-conquer approach, this also sets the stage for developing tessellation cases for higher order interpolants and the quadrilinear interpolant in four dimensions. We also demonstrate that apart from degenerate cases, Nielson's cases are, in fact, subsets of two basic configurations of the trilinear interpolant.

We propose an algorithm for the real time realistic simulation of multiple anisotropic scattering of light in a volume. Contrary to previous real-time methods we account for all kinds of light paths through the medium and preserve their anisotropic behavior.

Abstract This paper presents a method to accelerate algorithms that need a correct and complete visibility ordering of their data for rendering. The technique works by pre‐sorting primitives in object‐space using three lists (one for each axis: X, Y and Z), and then combining the lists using graphics hardware by rendering each list to a texture and merging the textures in the end. We validate our algorithm by applying it to the splatting technique using several types of rendering, including point‐based rendering and volume rendering. We also detail our hardware implementation for volume rendering using point sprites.

Abstract Images synthesized by light field rendering exhibit aliasing artifacts when the light field is undersampled; adding new light field samples improves the image quality and reduces aliasing but new samples are expensive to acquire. Light field rays are traditionally gathered directly from the source images, but new rays can also be inferred through geometry estimation. This paper describes a light field rendering approach based on this principle that estimates geometry from the set of source images using multi‐baseline stereo reconstruction to supplement the existing light field rays to meet the minimum sampling requirement. The rendering and reconstruction steps are computed over a set of planes in the scene volume, and output images are synthesized by compositing results from these planes together. The planes are each processed independently and the number of planes can be adjusted to scale the amount of computation to achieve the desired frame rate. The reconstruction fidelity (and by extension image quality) is improved by a library of matching templates to support matches along discontinuities in the image or geometry (e.g. object profiles and concavities). Given a set of silhouette images, the visual hull can be constructed and applied to further improve reconstruction by removing outlier matches. The algorithm is efficiently implemented by a set of image filter operations on commodity graphics hardware and achieves image synthesis at interactive rates.

VisIt is a richly featured visualization tool that is used to visualize some of the largest simulations ever run. The scale of these simulations requires that optimizations are incorporated into every operation VisIt performs. But the set of applicable optimizations that VisIt can perform is dependent on the types of operations being done. Complicating the issue, VisIt has a plugin capability that allows new, unforeseen components to be added, making it even harder to determine which optimizations can be applied. We introduce the concept of a contract to the standard data flow network design. This contract enables each component of the data flow network to modify the set of optimizations used. In addition, the contract allows for new components to be accommodated gracefully within VisIt's data flow network system.

We present the application of hardware accelerated volume rendering algorithms to the simulation of radiographs as an aid to scientists designing experiments, validating simulation codes, and understanding experimental data. The techniques presented take advantage of 32-bit floating point texture capabilities to obtain solutions to the radiative transport equation for X-rays. The hardware accelerated solutions are accurate enough to enable scientists to explore the experimental design space with greater efficiency than the methods currently in use. An unsorted hexahedron projection algorithm is presented for curvilinear hexahedral meshes that produces simulated radiographs in the absorption-only regime. A sorted tetrahedral projection algorithm is presented that simulates radiographs of emissive materials. We apply the tetrahedral projection algorithm to the simulation of experimental diagnostics for inertial confinement fusion experiments on a laser at the University of Rochester.

The bump-mapping technique is often used to represent bumps on objects such as bark on trees and craters on the moon. In order to render shadows cast by bumps, the horizon map method was proposed. The horizon map is a table which has, for each of a small collection of azimuthal directions, slopes from each viewpoint on the bump map (height field) to the corresponding horizon point, which is the highest viewable point seen from that viewpoint. We propose a more precise method for rendering bumpmap shadows using a both a horizon map and a distance map, to take curvature of surfaces into consideration. The distance map is a table, which has, for each azimuthal direction, horizontal projected distances from each point of the bump map to its corresponding horizon point. The proposed method can render shadows efficiently by using programmable graphics hardware.

Projection methods for volume rendering unstructured data work by projecting, in visibility order, the polyhedral cells of the mesh onto the image plane, and incrementally compositing each cell's color and opacity into the final image. Normally, such methods require an algorithm to determine a visibility order of the cells. The Meshed Polyhedra Visibility Order (MPVO) algorithm can provide such an order for convex meshes by considering the implications of local ordering relations between cells sharing a common face. However, in nonconvex meshes, one must also consider ordering relations along viewing rays which cross empty space between cells. In order to include these relations, the algorithm described in this paper, the scanning exact meshed polyhedra visibility ordering (SXMPVO) algorithm, scan-converts the exterior faces of the mesh and saves the ray-face intersections in an A-Buffer data structure which is then used for retrieving the extra ordering relations. The image which SXMPVO produces is the same as would be produced by ordering the cells exactly, even though SXMPVO does not compute an exact visibility ordering. This is because the image resolution used for computing the visibility ordering relations is the same as that which is used for the actual volume rendering and we choose our A-Buffer rays at the same sample points that are used to establish a polygon's pixel coverage during hardware scan conversion. Thus, the algorithm is image-space correct. The SXMPVO algorithm has several desirable features; among them are speed, simplicity of implementation, and no extra (i.e., with respect to MPVO) preprocessing.

We discuss two volume rendering methods developed at Lawrence Livermore National Laboratory. The first, cell projection, renders the polygons in the projection of each cell. It requires a global visibility sort in order to composite the cells in back to front order, and we discuss several different algorithms for this sort. The second method uses regularly spaced slice planes perpendicular to the X, Y, or Z axes, which slice the cells into polygons. Both methods are supplemented with antialiasing techniques to deal with small cells that might fall between pixel samples or slice planes, and both have been parallelized.

We describe an interactive visualization and modeling program for the creation of protein structures "from scratch." The input to our program is an amino acid sequence - decoded from a gene - and a sequence of predicted secondary structure types for each amino acid - provided by external structure prediction programs. Our program can be used in the set-up phase of a protein structure prediction process; the structures created with it serve as input for a subsequent global internal energy minimization, or another method of protein structure prediction. Our program supports basic visualization methods for protein structures, interactive manipulation based on inverse kinematics, and visualization guides to aid a user in creating "good" initial structures.

We propose to look at light fields from a dual space point of view. The advantage, in addition to revealing some new insights, is a framework that combines the benefits of many existing works. Using the well known two-plane-parameterization, we derive the duality between the 4-D light field and the 3-D world space. In the dual light field, rays become hyper points. We introduce the concept of hyperline. Then, cameras appear as hyperlines - camera hyperline (CHL) - mostly heterogeneous in color; scene points also appear as hyperlines - geometry hyperline (GHL) - mostly homogeneous in color. CHL's and GHL's are independent. The existence of one does not require or replace the other. When both exist, they cross each other at the dual ray hyper points. Both CHL and GHL-based light field rendering results are presented.

Direct volume rendering based on projective methods works by projecting, in visibility order, the polyhedral cells of a mesh onto the image plane, and incrementally compositing the cell’s color and opacity into the final image. Crucial to this method is the computation of a visibility ordering of the cells. If the mesh is “well‐behaved” (acyclic and convex), then the MPVO method of Williams provides a very fast sorting algorithm; however, this method only computes an approximate ordering in general datasets, resulting in visual artifacts when rendered. A recent method of Silva et al. removed the assumption that the mesh is convex, by means of a sweep algorithm used in conjunction with the MPVO method; their algorithm is substantially faster than previous exact methods for general meshes. In this paper we propose a new technique, which we call BSP‐XMPVO, which is based on a fast and simple way of using binary space partitions on the boundary elements of the mesh to augment the ordering produced by MPVO. Our results are shown to be orders of magnitude better than previous exact methods of sorting cells.

Selective visualization is a solution for visualizing data of large size and dimensionality. In this paper a new method is proposed for effectively rendering certain chosen parts among the full set of data in terms of a colour buffer, referred to as the virtual plane, for storing intermediate results. By this method, scientists may concentrate their attention on the contents of data in which they are interested. Besides, the method could be easily integrated with all the current direct volume rendering techniques, especially progressive refinement methods and selective methods. Copyright © 1999 John Wiley & Sons, Ltd.

A computer animated movie was produced, illustrating both 2D and 3D Hilbert curves, and showing the transition from 2D to 3D with the help of volume rendering.

This paper describes a volume rendering system for unstructured data, especially finite element data, that creates images with very high accuracy. The system will currently handle meshes whose cells are either linear or quadratic tetrahedra. Compromises or approximations are not introduced for the sake of efficiency. Whenever possible, exact mathematical solutions for the radiance integrals involved and for interpolation are used. The system will also handle meshes with mixed cell types: tetrahedra, bricks, prisms, wedges, and pyramids, but not with high accuracy. Accurate semi-transparent shaded isosurfaces may be embedded in the volume rendering. For very small cells, subpixel accumulation by splatting is used to avoid sampling error. A revision to an existing accurate visibility ordering algorithm is described, which includes a correction and a method for dramatically increasing its efficiency. Finally, hardware assisted projection and compositing are extended from tetrahedra to arbitrary convex polyhedra.

Flow volumes are extended for use in unsteady (time-dependent) flows. The resulting unsteady flow volumes are the 3D analogs of streaklines. There are few examples where methods other than particle tracing have been used to visualize time-varying flows. Since particle paths can become convoluted in time, there are additional considerations to be made when extending any visualization technique to unsteady flows. We present some solutions to the problems which occur in subdivision, rendering and system design. We apply the unsteady flow volumes to a variety of field types, including moving multi-zoned curvilinear grids.

The hemicube estimates of form factors are based on a finite set of sample directions. We obtain several optimal arrangements of sample directions, which minimize the variance of these estimates. They are based on changing the size or shape of the pixels or the shape of the hemicube, or using non-uniform pixel grids. The best reduces the variance by 43%. The variance calculation is based on the assumption that the errors in the estimate are caused by the projections of single polygon edges, and that the positions and orientations of these edges are random. This replaces the infinite dimensional space of possible environments by the two dimensional space of great circles on the unit sphere, making the numerical variance minimization possible.>

This tutorial survey paper reviews several different models for light interaction with volume densities of absorbing, glowing, reflecting, and/or scattering material. They are, in order of increasing realism, absorption only, emission only, emission and absorption combined, single scattering of external illumination without shadows, single scattering with shadows, and multiple scattering. For each model the paper provides the physical assumptions, describes the applications for which it is appropriate, derives the differential or integral equations for light transport, presents calculation methods for solving them, and shows output images for a data set representing a cloud. Special attention is given to calculation methods for the multiple scattering model.>

Computer animation of outdoor scenes is enhanced by realistic clouds. I discuss several different modelling and rendering schemes for clouds, and show how they evolved in my animation work. These include transparency-textured clouds on a 2D plane, smooth shaded or textured 3D cloud surfaces, and 3D volume rendering. For the volume rendering, I present various illumination schemes, including the density emitter, single scattering, and multiple scattering models.>

Vector field rendering is difficult in 3D because the vector icons overlap and hide each other. We propose four different techniques for visualizing vector fields only near surfaces. The first uses motion blurred particles in a thickened region around the surface. The second uses a voxel grid to contain integral curves of the vector field. The third uses many antialiased lines through the surface, and the fourth uses hairs sprouting from the surface and then bending in the direction of the vector field. All the methods use the graphics pipeline, allowing real time rotation and interaction, and the first two methods can animate the texture to move in the flow determined by the velocity field.>

Simulation of complex 3-dimensional phenomena generate data sets which are hard to comprehend using conventional 2-dimensionally oriented visualization tools. One way to overcome this limitation is to employ various volume visualization techniques. While early volume visualization techniques worked well on simple scalar volumes they failed to exploit frame buffer hardware capabilities and handle higher order data such as vector fields. Work at Lawrence Livermore National Laboratory has centered on developing new techniques and extending existing techniques. This paper describes the various algorithms developed for volume rendering, and presents new methods of examining vector fields in a volumetric fashion.

A method is described for switching smoothly between rendering algorithms as required by the amount of visible surface detail. The result will be more realism with less computation for displaying objects whose surface detail can be described by one or more bump maps. The three rendering algorithms considered are a BRDF, bump mapping, and displacement mapping. The bump mapping has been modified to make it consistent with the other two. For a given viewpoint, one of these algorithms will show a better trade-off between quality, computation time, and aliasing than the other two. The decision as to which algorithm is appropriate is a function of distance, viewing angle, and the frequency of bumps in the bump map.

Many of the current radiosity algorithms create a piecewise constant approximation to the actual radiosity. Through interpolation and extrapolation, a continuous solution is obtained. An accurate solution is found by increasing the number of patches which describe the scene. This has the effect of increasing the computation time as well as the memory requirements. By using techniques found in the finite element method, we can incorporate an interpolation function directly into our form factor computation. We can then use less elements to achieve a more accurate solution. Two algorithms, derived from the finite element method, are described and analyzed.

Volume visualization is becoming an important tool for understanding large 3D data sets. A popular technique for volume rendering is known as splatting. With new hardware architectures offering substantial improvements in the performance of rendering texture mapped objects, we present textured splats. An ideal reconstruction function for 3D signals is developed which can be used as a texture map for a splat. Extensions to the basic splatting technique are then developed to additionally represent vector fields.>

Flow volumes are the volumetric equivalent of stream lines. They provide more information about the vector field being visualized than do stream lines or ribbons. Presented is an efficient method for producing flow volumes, composed of transparently rendered tetrahedra, for use in an interactive system. The problems of rendering, subdivision, sorting, composing artifacts, and user interaction are dealt with. Efficiency comes from rendering only the volume of the smoke, and using hardware texturing and compositing.>

In order to visualize both clouds and wind in climate simulations, clouds were rendered using a 3D texture which was advected by the wind flow. The simulation is described. Rendering, the advection of texture coordinates, and haze effects are discussed. Results are presented.>

It is shown how data visualization fits into the broader process of scientific data analysis. Scientists from several disciplines were observed while they analyzed their own data. Examination of the observations exposed process elements outside conventional image viewing. For example, analysts queried for quantitative information, made a variety of comparisons, applied math, managed data, and kept records. The characterization of scientific data analysis reveals activity beyond that traditionally supported by computer. It offers an understanding which has the potential to be applied to many future designs, and suggests specific recommendations for improving the support of this important aspect of scientific computing.>

A computer was used to help study the packing of equal spheres in dimension four and higher. A candidate of the densest packing in 4-space is described. The configuration of 24 spheres touching a central sphere in this packing is shown to be rigid, unlike the analog in 3-space, in which the spheres can slide past each other. A system for interactively manipulating and visualizing such configurations is described. The Voronoi cell for a sphere is the set of points closer to its center than to any other sphere center in the packing. The packing density is the ratio of a sphere's volume to the average of the volumes of the Voronoi cells. A method of constructing Voronoi cells and computing their volumes that works in any dimension is presented. Examples of Voronoi cell volumes are given.>

A cone-sphere consists of two spheres, together with the part of the cylinder or cone tangent to the two spheres and lying between them. Cone-spheres can be rapidly rendered with shading, highlights, texture, or bump maps, and composited to create twisted or branched tubular structures.

The Torrance-Sparrow model for calculating bidirectional reflection functions contains a geometrical attenuation factor to account for shadowing and occlusions in a hypothetical distribution of grooves on a rough surface. Using an efficient table-based method for determining the shadows and occlusions, we calculate the geometric attenuation factor for surfaces defined by a specific table of bump heights.Diffuse and glossy specular reflection of the environment can be handled in a unified manner by using an integral of the bidirectional reflection function times the environmental illumination, over the hemisphere of solid angle above a surface. We present a method of estimating the integral, by expanding the bidirectional reflection coefficient in spherical harmonics, and show how the coefficients in this expansion can be determined efficiently by reorganizing our geometric attenuation calculation.

The shadow volume algorithm of Frank Crow was reorganized to provide information on the regions of illuminated space in front of each visible surface. This information is used to calculate the extra intensity due to atmospheric scattering, so when the atmosphere is partly in shadow, columns of scattered light will be visible. For efficiency in sorting the shadow edges, the image is computed in polar coordinates.

Algorithms are presented for raster and vector motion blur, which produce images and masks suitable for combination by the 21/2-D compositing process. The raster algorithm is based on a "skew, blur, unskew" scheme, using a very efficient one-dimensional blurring algorithm. The vector algorithm extends the ideas of anti-aliased scan conversion to motion blur.

This panel will discuss the issues and the problems associated with the simulation of natural phenomena. This is a difficult area of research since it generally involves complex data bases and in many instances time variant phenomena. The computational loads can become enormous as one considers the physics or the mathematical modeling of structures.

A ray-tracing procedural model is described, in which ocean waves and islands are rendered by different but related algorithms. The algorithms are based on analytic formulas involving arithmetic operations, trigonometric functions, and square roots, and are organized for a vectorizing compiler on a Cray 1, a “supercomputer” with a vector pipeline architecture. Height field methods are used, one vertical scan line at a time, to trace the direct rays to the ocean, where they are reflected. Approximate methods are then applied to find whether the reflected rays meet any other object on their way to the sky. The output, at eight bits per pixel, gives information for shading, e.g. the angle of the surface normal for rays meeting the islands, or the angle of elevation from the horizon for rays continuing unobstructed to the sky.

The optical printer can be considered as an optical analog computer, which can perform geometric transformations and simple arithmetic operations on pictures very efficiently. The principles of operation of the printer are explained, and many of its applications to computer animation are listed and discussed briefly.

The ATOMS program, written at Bell Telephone Laboratory, is capable of determining the visible portions of a scene consisting of interpenetrating spheres and cylinders, put together to represent “space-filling” or “ball-and-stick” molecular models. The Lawrence Livermore Laboratory version contains enhancements to add shading and highlights, and to render the spheres on film as ellipses, so they will appear round when projected in various wide-screen formats.

At Carnegie Mellon University, the PDP-10 and PDP-11 computers have been used to trace thin sections of stained neurons in a lobster swimmeret ganglion, and to reconstruct the global structure of the neuron from the sections. This paper will present two sorts of graphic pictures of the final reconstructed neuron; vector drawings with hidden lines, and shaded raster drawings of the neuron inside a transparent shell representing the outer surface of the ganglion. The hidden line program handles curving cylindrical tubes of varying radii, and shows only the visible contours. The raster program is a modification of Sproul's version of the Watkins Algorithm, and applies Phong shading to both opaque and semi-transparent surfaces. The shaded output is rendered on the Carnegie-Mellon video system.

In 1948, Stephen Smale, a mathematician who was then at the University of Chicago, proved that it was possible to turn the surface of a sphere inside out by a special kind of deformation called a “regular homotopy”. During a regular homotopy, the surface must remain continuous, without any tears or holes, and smooth, without any creases or singularities. However, the surface is allowed to cross and pass through itself. In other words, if we consider a position of the sphere as a function from the standard round spherical surface S2 into the three dimensional space R3, this function must be continuous and smooth (C1), but not necessarily one-to-one. Such a function is called an immersion, and some sample immersions of the sphere are shown in the figures here. A regular homotopy is then a continuous family of immersions, satisfying a technical condition that its partial derivatives with respect to position coordinates on S2 are continuous functions of time, as well as of position on S2. This is the smoothness condition which prohibits creases and kinks. Smale proved that there was a regular homotopy between any two immersions of the sphere. The ordinary round sphere and an inside-out round sphere are special cases of immersions, so there must be a regular homotopy between them. Smale’s proof was by induction and in principal contained a construction of an actual regular homotopy, but one so complicated that it would be impossible to visualize. Once the solution was known to be possible, a number of people tried to invent simple and easily visualized homotopies. One of these is discussed in a Scientific American article [7], which contains a more extensive description of the problem, and a complete sequence of illustrations for the homotopy. However, in order to understand the deformation, the reader must imagine the motions between the illustrations in the sequence, and convince himself that they are continuous and smooth. Clearly it would be preferable to have a movie showing the whole deformation. Since each frame is a different complicated surface, this is an obvious candidate for computer animation. The film projected at the meeting and illustrated in the figures here shows a regular homotopy developed by Bernard Morin, a blind French mathematician who was also instrumental in the creation of several other such homotopies. The one used here is among the simplest known, since it contains only fourteen critical stages, the minimum among the homotopies known at present. There were two main problems in the creation of the film: first the mathematical definition of the homotopy to the computer, and second, the efficient computation of the frames for output through suitable hardware onto film. These will be discussed in the sections to follow.