Jon G. Rokne

Double-step incremental generation of lines and circles

Abstract An explicit discussion of the possible speed-up of curve generation due to an increase in the size of increment has not appeared in literature. This paper proposes a methodology and its rationale for scan-converting lines and circles two pixels per iteration. The double-step line and circle algorithms require the same amount of integer arithmetic per iteration as the single-step algorithms but only half the number of iterations. The algorithms take advantage of some simple properties of discrete loci of mathematical curves in the raster plane.

Complex interval arithmetic

Complex interval arithmetic is defined using real interval arithmetic. Complex interval division is defined so as to assure smallest possible resulting intervals.

Newton's method under mild differentiability conditions with error analysis

SummaryThe concept of majorizing sequences introduced by Rheinboldt (SIAM J.N.A. 1968) is used to prove convergence for Newtons method for operator equations of the formT f=θ when the operator satisfied the condition that the Fréchet derivative is Hölder continuous.A detailed analysis of computational errors is given for Newtons method applied to operators with Hölder continuous derivatives. This analysis is shown to reduce the analysis of Lancaster (Num. Math. 1968) when the operator has a continuous second derivative.The above analysis is applied to an example of a second order differential equation.

Updating the topology of the dynamic Voronoi diagram for spheres in Euclidean d -dimensional space

The problem of dynamic maintenance of a Voronoi diagram for a set of spheres moving independently in d-dimensional space is addressed in this paper. The maintenance of this Voronoi diagram for spheres moving along given trajectories, requires the calculation of topological events, that occur when d + 2 spheres become tangent to a common sphere. The criterion for determination of the topological event in the Euclidean metric is derived as a solution of a system of non-linear algebraic equations. The criterion is given in the form of polynomial algebraic equations dependent on the coordinates and trajectories of the moving spheres. These equations are solved using numerical methods. Application of the method to study the structure of a system of polydisperse spheres in a three-dimensional Euclidean space is briefly discussed.

An Algorithmic Reflectance and Transmittance Model for Plant Tissue

Recent developments in rendering have provided very realistic images. However, these images rarely show organic objects. We believe that one of the main difficulties of rendering these objects realistically is the lack of reflectance and transmittance models oriented to organic materials. In this paper an algorithmic reflectance and transmittance model for plant tissue oriented to computer graphics applications is presented. The model accounts for the three components of light propagation in plant tissues, namely surface reflectance, subsurface reflectance and transmittance, and mechanisms of light absorption by pigments present in these tissues. The model design is based on the available biological information, and it is controlled by a small number of biologically meaningful parameters. Its formulation, based on standard Monte Carlo techniques, guarantees its easy incorporation into most rendering systems. The spectral curves of reflectance and transmittance computed by the model are compared with measured curves from actual experiments.

Fast line scan-conversion

A major bottleneck in many graphics displays is the time required to scan-convert straight line segments. Most manufacturers use hardware based on Bresenhams [5] line algorithm. In this paper an algorithm is developed based on the original Bresenham scan-conversion together with the symmetry first noted by Gardner [18] and a recent double-step technique [31]. This results in a speed-up of scan-conversion by a factor of approximately 4 as compared to the original Bresenham algorithm. Hardware implementations are simple and efficient since the property of using only shift and increment operations is preserved.

ONE-DIMENSIONAL THERMAL SHOCK PROBLEM WITH TWO RELAXATION TIMES

The theory of generalized thermoelasticity with two relaxation times is used to solve a boundary value problem of an isotropic elastic half-space with its plane boundary either held rigidly fixed or stress-free and subjected to a sudden temperature increase. An approximate small-time solution is obtained by using the Laplace transform method. Numerical values of displacement, stress, and temperature are obtained and displayed graphically. Two discontinuities in both the displacement and temperature functions and that the stress has infinite discontinuities at these two wave fronts were noted

Virtual spectrophotometric measurements for biologically and physically-based rendering

The group of measurements necessary to characterize both the color and surface finish of an object is called the measurement of appearance of an object [4]. This group of measurements involves the spectral energy distribution of propagated light, measured in terms of reflectance and transmittance, and the spatial distribution of that light, measured in terms of the bidirectional reflectance distribution function (BRDF) and the bidirectional transmittance distribution function (BTDF). The variations in the spectral energy distribution affect appearance characteristics such as hue, lightness and saturation, while the changes in the spatial distribution affect appearance characteristics such as gloss, reflection haze, transmission haze, luster and translucency as noted by Hunter and Harold [4]. Measuring these appearance characteristics is crucial for realistic rendering.

Bounds for an interval polynomial

We discuss the evaluation of the range of values of an interval polynomial over an interval. Several algorithms are proposed and tested on numerical examples. The algorithms are based on ideas by Cargo and Shiska [2] and Rivlin [4]. The one basic algorithm uses Bernstein polynomials. It is shown to converge to the exact bounds and it has furthermore the property that if the maximum respectively the minimum of the polynomials occurs at an endpoint of the interval then the bound is exact. This is a useful property in routines for polynomials zeros. The other basic method is based on the meanvalue theorem and it has the advantage that the degree of approximation required for a certain apriori tolerance is smaller than the degree required in the Bernstein polynomial case. The mean value method is shown to be at least quadratically convergent and the Bernstein polynomial method is shown to be at least linearly convergent.ZusammenfassungIn dieser Arbeit wird die Auswertung von Schranken für den Wertebereich für Intervallpolynome über einem Intervall diskutiert. Mehrere Algorithmen sind vorgeschlagen und sie sind an numerischen Beispielen ausgewertet. Die Algorithmen basieren auf Ideen von Cargo und Shiska [2] und Rivlin [4]. Der eine fundamentale Algorithmus benutzt Polynome, und die Konvergenz gegen die genauen Schranken wird für diesen Algorithmus gezeigt. Ferner ist es von Nutzen, daß, wenn die obere bzw. untere Schranke des Polynoms am Ende des Intervalles auftritt, die Schranke genau ausgewertet wird. Die andere Methode basiert auf dem Mittelwertsatz. Diese Methode hat den Vorteil, daß der Grad der Approximation, der von einer gewissen Toleranz verlangt wird, kleiner ist als im Fall des Bernstein-Polynoms. Es wird gezeigt, daß die Mittelwert.-satzmethode mindestens quadratisch konvergent ist und daß die Bernstein-Polynom-Methode mindestens linear konvergent ist.

Disk Be´zier curves

Disk Bezier curves are defined. Properties of these curves discussed are algorithms, a centered form, their use in approximating a real curve, their envelope and an approximation to the curves. Ball Bezier curves in three dimensions are introduced and some comments on the generalization of the results for disk Bezier curves to ball Bezier curves are given.