Robert P. Burton

A Hidden-Line Algorithm for Hyperspace

An object-space hidden-line algorithm for higher-dimensional scenes has been designed and implemented. Scenes consist of convex hulls of any dimension, each of which is compared against the edges of all convex hulls not eliminated by a hyperdimensional clipper, a depth test after sorting and a minimax text.Hidden and visible elements are determined in accordance with the dimensionality of the selected viewing hyperspace. When shape alone is the attribute of interest, hidden-line elimination need be performed only in that hyperspace.The algorithm is of value in the production of shadows of hyperdimensional models, including but not limited to four-dimensional space-time models, the hyperdimensional elementary catastrophe models and multivariate statistical models.

The Need for an Industry Standard of Accuracy for Elementary-Function Programs

In recent years there has been some concern for the lack of accuracy in mathematical software provided by computer manufacturers. Of particular concern is the inaccuracy of the subprograms for the elementary mathematical functions. These functions are used extensively in many applications and must be both accurate and efficient. In response to user concern in the late 1960s and early 1970s, several companies made significant improvements in the accuracy of their e lementary-funct ion libraries [3, 4, 8]. But now, more than a decade later, the computer industry as a whole has not effected any consequential changes in, nor has it adopted a s tandard

Generation and graphical analysis of Mandelbrot and Julia sets in more than four dimensions

Abstract A method to define and generate Mandelbrot and Julia sets in more than four dimensions is presented. A doubling process is used to create from the set of real numbers a hypercomplex number system of arbitrary dimension. Since the new number system is closed under addition and multiplication, it can be used to generate Mandelbrot and Julia sets of corresponding dimension. Generation of these sets in more than four dimensions is discussed. A graphical analysis manifests the sets are fractal in these higher dimensions. Symmetrical properties of Mandelbrot and Julia sets are observed and reported.

Extension of star coordinates into three dimensions

Traditional Star Coordinates displays a multi-variate data set by mapping it to two Cartesian dimensions. This technique facilitates cluster discovery and multi-variate analysis, but binding to two dimensions hides features of the data. Three-dimensional Star Coordinates spreads out data elements to reveal features. This allows the user more intuitive freedom to explore and process the data sets. Three-dimensional Star Coordinates is implemented by extending the data structures and transformation facilities of traditional Star Coordinates. We have given high priority to maintaining the simple, traditional interface. We simultaneously extend existing features, such as scaling of axes, and add new features, such as system rotation in three dimensions. These extensions and additions enhance data visualization and cluster discovery. We use three examples to demonstrate the advantage of three-dimensional Star Coordinates over the traditional system. First, in an analysis of customer churn data, system rotation in three dimensions gives the user new insight into the data. Second, in cluster discovery of car data, the additional dimension allows the true shape of the data to be seen more easily. Third, in a multi-variate analysis of cities, the perception of depth increases the degree to which multi-variate analysis can occur.

Characterization and categorization of higher-dimensional presentation techniques

Existing, documented techniques for the presentation of higher dimensionaL information are characterized. Techniques include: Symbolic Star PLots, Chernoff Faces, Gtyphs, Boxes, Profile PLots, SymboLic Scatter PLots, KLeiner-Hartigan Tree SymboLs, GeneraLized Draftsman DispLays, Andrews PLots, ParaLLeL Axes Graphics, and Cartesian Hyperspace Graphics. Each technique is evaLuated based on accuracy, simpLicity, clarity, appearance, well-designed structure, information leveL, dimansonaL capacity, flexibility, interpretability, visual impact, mastery time, and computational tractability. Strengths, weaknesses and applicabilities of each technique are determined. Techniques are categorized as symbolic and non-symbolic. Characteristics of each category are identified.

A curious recursive function

The function defined recursively by h(0) = 0h(k) = k − h(h(k − 1)) is investigated. It is shown that h has the closed form solution h(k) = floor((π − 1)(k + 1)) where π is the positive root of x(x − 1) = 1 (the golden mean). Its closed form solution is similar to the closed form solution of the winning strategy for a two player nim-like game that was popular in student lounges in the early seventies. The solution of the recursive equation depends on arithmetic accidents involving the rationality and irrationality of various quantities associated with a quadratic equation.

First university course in computer graphics

Abstract A university level, fourth generation, first course in computer graphics has been designed and presented in three formats: • • over a full semester at a major private university • • over a full semester on site in a corporate setting • • during a single, intensive week in a corporate setting In all three formats, course content has included a history of computer graphics, output primitives, 2D and 3D transformations, 3D concepts and representations, windowing and clipping, 3D viewing, fractals, sweep representations, constructive solid geometry, quadtrees and octrees, hidden-element removal, modelling and displaying light intensities, surface shading, segmentation, modelling, curved surfaces, display devices and hard copy devices, graphics workstations, display processors, graphics software and standards, interactive input devices and techniques, input functions, color models and user interface design. The first two presentation formats have included a major graphics software system development requirement including polygon management, transformation routines, the viewing pipeline, clipping, scan conversion, hidden surface removal and one or more student-selectable options. The third format has permitted minor laboratory experience with a developed graphics software system.

Multivariate data visualization via outdoor scenes

Visualization of multivariate data presents a challenge due to the sheer dimensionality and density of information. When presenting the data symbolically, this high information dimensionality and density makes it difficult to develop a symbology capable of displaying it in a single presentation. One approach to multivariate visualization involves creating symbols with higher dimensionality. Higher dimensional symbols can be problematic, since they typically require significant human attentive processing to interpret, offsetting their greater informational capacity. Although attempts have been made to develop higher-dimensional symbols that are processed in a preattentive fashion, success has proven elusive. Recent cognitive research indicates that outdoor scenes are processed in a preattentive manner. We evaluate outdoor scenes as a candidate for developing an effective higher-dimensional symbology by generating proof-of-concept images and comparing them to related methods.

Versatile reactive navigation

Most autonomous mobile agents operate in a highly constrained environment. Despite significant research, existing solutions are limited in their ability to handle heterogeneous constraints within highly dynamic or uncertain environments. This paper presents a novel maneuver selection technique suited for both 2D and 3D environments with highly dynamic maneuvering constraints and multiple mobile obstacles. Agents may have any arbitrary set of nonholonomic control variables; maneuvers can be constrained by a broad class of function inequalities, including time-dependent constraints involving nonlinear relationships between controlled and agent-state variables. The resulting algorithm has been implemented to run in real time using only a fraction of the CPUs capacity on an ordinary notebook computer and performs well in a number of taxing simulated situations.

Piles for hyperdimensional graphics

Abstract Traditional 2-D symbolic hyperdimensional presentation techniques are deficient for comparing corresponding values on different symbols. This paper presents piles which overcome this problem by stacking the synbols on an axis. Columns, a piled technique based on hyperdimensional stars, are presented as an illustration. An application to the federal tax system demonstrates the analysis of data using this technique and demonstrates its superiority to traditional 2-D techniques.