Alan Norton

Generation and display of geometric fractals in 3-D

We present some straightforward algorithms for the generation and display in 3-D of fractal shapes. These techniques are very general and particularly adapted to shapes which are much more costly to generate than to display, such as those fractal surfaces defined by iteration of algebraic transformations. In order to deal with the large space and time requirements of calculating these shapes, we introduce a boundary-tracking algorithm particularly adapted for array-processor implementation. The resulting surfaces are then shaded and displayed using z-buffer type algorithms. A new class of displayable geometric objects, with great diversity of form and texture, is introduced by these techniques.

Interactive simulation in a multi-person virtual world

A multi-user Virtual World has been implemented combining a flexible-object simulator with a multisensory user interface, including hand motion and gestures, speech input and output, sound output, and 3-D stereoscopic graphics with head-motion parallax. The implementation is based on a distributed client/server architecture with a centralized Dialogue Manager. The simulator is inserted into the Virtual World as a server. A discipline for writing interaction dialogues provides a clear conceptual hierarchy and the encapsulation of state. This hierarchy facilitates the creation of alternative interaction scenarios and shared multiuser environment.

Clamping: A method of antialiasing textured surfaces by bandwidth limiting in object space

An object space method is given for interpolating between sampled and locally averaged signals, resulting in an antialiasing filter which provides a continuous transition from a sampled signal to its selectively dampened local averages. This method is applied to the three standard Euclidean dimensions and time, resulting in spatial and frame to frame coherence. The theory allows filtering of a variety of functions, including continuous and discrete representations of planar texture.

Julia sets in the quaternions

Abstract Recent mathematical work on the dynamics of complex analytic functions has given rise to a new subject matter for computer graphics. The combination of mathematical theory and computer graphics has resulted in new insight into the nature of some of the simplest of mathematical objects. second-degree polynomials. Most of that work has focused on the possibilities within the two-dimensional complex plane. This article shows how these investigations may be extended to higher dimensions, resulting in fractals that naturally reside in the 4-dimensional quaternions. Particular attention is paid to the formula ax2 + b. A method is given for obtaining various interconnection patterns for the Julia sets in 4-space, and the results are displayed in 3-D computer graphics.

Parallel Quicksort using fetch-and-add

A parallelization of the Quicksort algorithm that is suitable for execution on a shared memory multiprocessor with an efficient implementation of the fetch-and-add operation is presented. The partitioning phase of Quicksort, which has been considered a serial bottleneck, is cooperatively executed in parallel by many processors through the use of fetch-and-add. The parallel algorithm maintains the in-place nature of Quicksort, thereby allowing internal sorting of large arrays. A class of fetch-and-add-based algorithms for dynamically scheduling processors to subproblems is presented. Adaptive scheduling algorithms in this class have low overhead and achieve effective processor load balancing. The basic algorithm is shown to execute in an average of O(log(N)) time on an N-processor PRAM (parallel random-access machine) assuming a constant-time fetch-and-add. Estimated speedups, based on simulations, are also presented for cases when the number of items to be sorted is much greater than the number of processors. >

Modelling physical objects for simulation

The authors present a framework for modeling and simulating objects with physical attributes. Rather than attaching physical properties to geometric shapes, one directly designs and manipulates objects with intrinsic dynamic properties such as stiffness, mass, and angular momentum. Flexible and rigid bodies can interact in the same simulated physical environment, responding to collisions, fluid velocity fields, friction, and gravity. A modeler and simulator organized along these principles has been implemented and used in computer animation. The results obtained with these systems are presented.<<ETX>>

Aspects of motion design physically-based animation

We explore ways in which physically-based simulation can generate the motion of objects for computer animation. We model flexible and brittle objects and their interaction with wind fields using classical mechanics. We have enhanced our animation environment with a capability that allows quick visual preview of simulations. This is an essential tool for prototyping motion and previewing scenes before they are converted into high quality images with a raytracer. Complex motion can be both specified and controlled by designing wind fields and by making use of the preview capability. Examples are given of how fields and preview are used to create animated scenes that involve hundreds of objects in wind fields.

Use of curves in rendering fractures

Recently, physically-based models have been used to simulate fractures in solid objects. Due to the finite grid used in this type of simulation, aliasing artifacts occur at fracture edges revealing the structure of the grid. We have developed a method for identifying the fractured edges based on the specific grid structure of the model. We then use arbitrary order Bezier curves to modify the grid structure, concealing it at fracture edges. This results in a more convincing fracture edge.

Chapter 33 – Julia sets in the quaternions

Publisher Summary Several studies have appeared in recent years demonstrating the visual effects obtainable from applying 2Dcomputer graphics to complex polynomials. Fascinating as such pictures are, they are only slim fragments compared to the 3D or 4D physical reality. Quadratics reside in higher dimensions, and this chapter presents one such extension––that is, the usage of 4D quaternion algebra to define structures possessing complex patterns of infinitely repeating geometric structure. Recent mathematical work on the dynamics of complex analytic functions has given rise to a new subject matter for computer graphics. The combination of mathematical theory and computer graphics has resulted in new insight into the nature of some of the simplest of mathematical objects: second-degree polynomials. Most of that work has focused on the possibilities within the 2D complex plane. This chapter shows the extension of these investigations to higher dimensions, thereby resulting in fractals that naturally reside in the 4D quaternions. Particular attention is paid to the formula ax2 + b. A method is given for obtaining various interconnection patterns for the Julia sets in 4-space, and the results are displayed in 3D computer graphics.